simple geometric model

:EXPORT_FILE_NAME: simple_geometric_model

description of all variables

labeled_cartoon.png
\begin{equation} F=\sigma\left(S_{0}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation}
description value units reference type python variable associated equation LaTeX variable
invagination height   \(nm\)   input (variable) H_i   \(H_{i}\)
membrane tension 0.001 \(pN/nm\) (Hassinger et al. 2017) fixed sigma   \(\sigma\)
total plane radius 2000 \(nm\)   fixed A_t   \(A_{t}\)
final vesicle radius 50 \(nm\) (Kaksonen and Roux 2018) fixed vesicle_radius   \(R_{v}\)
coat bending rigidity 160 \(pN nm\) (Dmitrieff and Nédélec 2015) fixed K   \(K\)
spontaneous curvature 1/50 \(nm^{-1}\) (Hassinger et al. 2017) fixed C_i   \(C_{i}\)
droplet-cytosol surface tension 0.005 \(pN/nm\) (Jawerth et al. 2018) fixed gamma_d   \(\gamma_{d}\)
droplet-membrane wetting tension 0.002 \(pN/nm\)   fixed gamma_m   \(\gamma_{m}\)
initial droplet plane radius       fixed      
droplet-invagination surface tension 0.002 \(pN/nm\)   fixed gamma_i   \(\gamma_{i}\)
turgor pressure 0.1 \(pN/nm^2\) (Ma and Berro 2019) fixed P   \(P\)
relative invagination height   unitless   input (variable) h rel-h \(h\)
droplet volume   \(nm^3\)   intermediate output V_d   \(V_{d}\)
invagination surface area   \(nm^2\)   intermediate output S_i invagination-area \(S_{i}\)
initial droplet spherical radius 600 \(nm\)   intermediate output drop_radius    
droplet-flat membrane contact area   \(nm^2\)   intermediate output S_m md-area \(S_{m}\)
invagination spherical radius   \(nm\)   intermediate output R_i invagination-radius \(R_{i}\)
droplet-cytosol contact area   \(nm^2\)   intermediate output S_d dc-contact-area \(S_{d}\)
droplet contact angle   rad   intermediate output theta_d droplet-contact-angle \(\theta_{d}\)
invagination contact angle   rad   intermediate output theta_i invagination-angle \(\theta_{i}\)
invagination plane radius   \(nm\)   intermediate output A_i invagination-plane-radius \(A_{i}\)
droplet + invagination plane radius   \(nm\)   intermediate output A_d droplet-plane-radius \(A_{d}\)
outside membrane surface area   \(nm^2\)   intermediate output S_o outside-membrane-area \(S_{o}\)
invagination volume   \(nm^3\)   intermediate output V_i invagination-volume \(V_{i}\)
droplet + invagination volume   \(nm^3\)   intermediate output V_b di-volume \(V_{b}\)
droplet + invagination height   \(nm\)   intermediate output H_d di-height \(H_{d}\)
droplet + invagination spherical radius   \(nm\)   intermediate output R_d di-cap-radius \(R_{d}\)
free energy   \(pN \cdot nm\)   output F f-forces f-shapes \(F\)
membrane tension energy   \(pN \cdot nm\)   output membrane_tension    
bending energy   \(pN \cdot nm\)   output bending_energy    
surface tension energy   \(pN \cdot nm\)   output surface_tension    
wetting energy   \(pN \cdot nm\)   output wetting_energy    
turgor pressure energy   \(pN \cdot nm\)   output turgor_pressure    
membrane curvature   \(nm^{-1}\)   output 2÷R_i    

4.114 \(pN \cdot nm\) = 1 \(k_{B}T\) (“kT (Energy)” 2021)

references

Dmitrieff, Serge, and François Nédélec. 2015. “Membrane Mechanics of Endocytosis in Cells with Turgor.” PLoS Computational Biology 11 (10). doi:10.1371/journal.pcbi.1004538.
Hassinger, Julian E., George Oster, David G. Drubin, and Padmini Rangamani. 2017. “Design Principles for Robust Vesiculation in Clathrin-Mediated Endocytosis.” Proceedings of the National Academy of Sciences, January, 201617705. doi:10.1073/pnas.1617705114.
Jawerth, Louise M., Mahdiye Ijavi, Martine Ruer, Shambaditya Saha, Marcus Jahnel, Anthony A. Hyman, Frank Jülicher, and Elisabeth Fischer-Friedrich. 2018. “Salt-Dependent Rheology and Surface Tension of Protein Condensates Using Optical Traps.” Physical Review Letters 121 (25): 258101. doi:10.1103/PhysRevLett.121.258101.
Kaksonen, Marko, and Aurélien Roux. 2018. “Mechanisms of Clathrin-Mediated Endocytosis.” Nature Reviews Molecular Cell Biology, February. doi:10.1038/nrm.2017.132.
“kT (Energy).” 2021. Wikipedia, January.
Ma, Rui, and Julien Berro. 2019. “Mechanics of Membrane Internalization against Osmotic Pressure: Boundary Conditions Make a Difference.” Biorxiv, February, 558890. doi:10.1101/558890.
“Python - Spherical Coordinates Plot in Matplotlib.” n.d. Stack Overflow. https://stackoverflow.com/questions/36816537/spherical-coordinates-plot-in-matplotlib.
NO_ITEM_DATA:ma_endocytosis_2019

equations

summary

labeled_cartoon.png

arranged by force contributions: 

F = membrane tension + bending energy + droplet-cytosol surface tension -
droplet-membrane surface tension - droplet-invagination surface tension + turgor
pressure
\begin{equation} \label{ F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation} \begin{equation} F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation}

rearranged by shapes:

\begin{equation} F= \sigma S_{o} + \left(\sigma - \gamma_{m} \right) S_{m} + \gamma_{d} S_{d} + \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} + P V_{i} + \left(\sigma - \gamma_{i} \right) S_{i} \end{equation}

broken down to shape functions and constant \(C\):

\begin{equation} F(\Delta S_{o}, \Delta S_{m}, S_{d}, R_{i}, V_{i})= F_{1}(\Delta S_{o}) + F_{2}(\Delta S_{m}) + F_{3}(S_{d}) + F_{4}(R_{i}) + F_{5}(V_{i}) + C \end{equation} \begin{equation} F_{1}(\Delta S_{o}) = \sigma \Delta S_{o} \end{equation} \begin{equation} \Delta S_{o}=\alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \label{alpha} \alpha = - \left(\frac{\pi\left(\frac{\gamma_{m}}{\gamma_{d}}+1\right)^{3}} {\left(-\frac{\gamma_{m}}{\gamma_{d}}+1\right)\left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \label{} F_{2}(\Delta S_{m}) = \left(\sigma - \gamma_{m} \right) \Delta S_{m} \end{equation} \begin{equation} \Delta S_{m}= - \alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \label{} F_{3}(S_{d}) = \gamma_{d} S_{d} \end{equation} \begin{equation} S_{d}=\epsilon \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \epsilon = \left(\frac{8 \pi}{\left(-\frac{\gamma_{m}}{\gamma_{d}}+1\right) \left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \label{} \begin{array}{l} F_{4}(R_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} \end{array} \end{equation} \begin{equation} R_{i} = \frac{\sqrt{S_i}}{2 \sqrt{\pi } h} \end{equation} \begin{equation} \label{} F_{4}(h) = 2 K \left(C_i \sqrt{S_i}-2 \sqrt{\pi } h\right)^{2} \end{equation} \begin{equation} \label{} F_{5}(V_{i}) = P V_{i} \end{equation} \begin{equation} \label{} V_{i}=\eta \left(3 h-2 h^{3}\right) \end{equation} \begin{equation} \label{} \eta=\frac{S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} \begin{equation} \label{} C = \sigma \pi A_t^2+S_i \left(\gamma _m-\gamma _i\right) \end{equation}

all combined into full free energy equation:

\begin{equation} F(h)=\left(\gamma_{m} \alpha+\gamma_{d} \epsilon\right)\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3}+\left(\sigma-\gamma_{m}\right) S_{i} h^{2} +2 K\left(C_{i} \sqrt{S}_{i}-2 \sqrt{\pi} h\right)^{2}+P \eta \left(3 h-2 h^{3}\right)+C \end{equation}

expanded polynomial:

\begin{equation} F(h)= \left(\gamma_{m} \alpha+\gamma_{d} \epsilon\right) \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} - 2 P \eta h^{3}+\left(\left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi\right) h^{2} +\left(3 P \eta-8 K C_{i} \sqrt{\pi S_{i}}\right) h+B \end{equation} \begin{equation} B=C+2 K C_{i}^{2} S_{i} \end{equation}

individual shape solutions

relative invagination height

h.jpg
\begin{equation} \label{rel-h} h = \frac{H_{i}}{\left(\frac{S_{i}}{\pi}\right)^{1/2}} \end{equation} \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation}

droplet-membrane contact angle

2020-09-21_19-17-56_screenshot.png
Figure 1: contact angle cartoon contact-angle
\begin{equation} \begin{array}{l} \gamma_{S G}=\gamma_{S L}+\gamma_{L G} \cos \theta \\ \sigma+\gamma_{m}=\sigma+\gamma_{d} \cos \theta_{d} \\ \cos \theta_{d}=\frac{\gamma_{m}}{\gamma_{d}} \\ \sin \theta_{d}=\sqrt{1 - \cos \theta_{d}^{2}} \\ \sin \theta_{d}=\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}} \end{array} \end{equation} \begin{equation} \label{droplet-contact-angle} \theta_{d}=\arccos \left(\frac{\gamma_{m}}{\gamma_{d}}\right) \end{equation}

outside membrane area

\begin{equation} \label{outside-membrane-area} S_{0}=\pi\left(A_{t}^{2}-A_{d}^{2}\right) \end{equation}

circular plane radii

\begin{equation} \begin{array}{l} \sin \theta=\frac{a}{r} \\ a=r \sin \theta} \\ A_{i}=R_{i} \sin \theta_{i} \end{array} \end{equation} \begin{equation} \label{droplet-plane-radius} A_{d}=R_{d} \sin \theta_{d} \end{equation}

droplet-cytosol contact area

\begin{equation} \label{dc-contact-area} \begin{array}{l} A=2 \pi r^{2}(1-\cos \theta) \\ S_{d}=2 \pi R_{d}^{2}\left(1-\cos \theta_{d}\right) \end{array} \end{equation}

invagination surface area

\begin{equation} \label{invagination-area} S_{i}=4 \pi R_{v}^{2} \end{equation}

invagination volume

\begin{equation} \label{invagination-volume-a-h} \begin{array}{l} V=\frac{1}{6} \pi h\left(3 a^{2}+h^{2}\right) \\ V_{i}=\frac{1}{6} \pi H_{i}\left(3 A_{i}^{2}+H_{i}^{2}\right) \end{array} \end{equation}

alternately:

\begin{equation} \label{invagination-volume} \begin{array}{l} V=\frac{\pi}{3} h^{2} \left(3 r - h \right) \\ V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{array} \end{equation}

invagination plane radius

\begin{equation} \begin{array}{l} A=\pi\left(a^{2}+h^{2}\right) \\ S_{i}=\pi\left(A_{i}^{2}+H_{i}^{2}\right) \\ A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}} \end{array} \end{equation} \begin{equation} \label{invagination-plane-radius} A_{i}=\sqrt{\frac{S_{i}}{\pi} \left(1 - h^{2}\right)} \end{equation}

invagination radius

\begin{equation} \label{invagination-radius} \begin{aligned} A &=2 \pi r h \\ S_{i} &=2 \pi R_{i} H_{i} \\ R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{aligned} \end{equation}

invagination contact angle

\begin{equation} \label{invagination-angle} \begin{array}{l} S_{i}=2 \pi R_{i}^{2}\left(1-\cos \theta_{i}\right) \\ \theta_{i}=\arccos \left(1-\frac{S_{i}}{2 \pi R_{i}^{2}}\right) \end{array} \end{equation}

TODO droplet volume

  • should depend on initial plane radius

    \begin{equation} \label{d-volume} V_{b}=V_{d}+V_{i} \end{equation}

droplet + invagination volume

\begin{equation} \label{di-volume} V_{b}=V_{d}+V_{i} \end{equation}

droplet + invagination spherical cap radius

\begin{equation} \label{di-cap-radius} \begin{array}{l} V=\frac{\pi}{3} r^{3}(2+\cos \theta)(1-\cos \theta)^{2} \\ V_{b}=\frac{\pi}{3} R_{d}^{3}\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2} \\ R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \end{array} \end{equation}

droplet + invagination height

\begin{equation} \label{di-height} \begin{array}{l} A=2\pi r h \\ S_{d}=2\pi R_{d} H_{d} \\ H_{d}=\frac{S_{d}}{2 \pi R_{d}} \end{array} \end{equation}

flat membrane-droplet contact area

\begin{equation} \label{md-area} S_{m}=\pi\left(A_{d}^{2}-A_{i}^{2}\right) \end{equation}

separate force components (original version)

\begin{equation} \label{f-forces} F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation}

fully substituted

\begin{equation} \label{free-energy} F=\sigma(\pi(A_{t}^{2}-(((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2})+(\pi((((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2}-(\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}))+(4 \pi R_{v}^{2}))+ \frac{K}{2} (4 \pi R_{v}^{2})(\frac{2}{(\frac{(4 \pi R_{v}^{2})}{2 \pi \left(\frac{S_{i} h}{\pi}\right)})}-2C_{i})^{2}+ \gamma_{d} (2 \pi R_{d}^{2}(1-(\frac{\gamma_{m}}{\gamma_{d}})))- \gamma_{m} (\pi((((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2}-(\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}))- \gamma_{i} (4 \pi R_{v}^{2})+ P (\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})) \end{equation}

separate shape force components

\begin{equation} F(\Delta S_{o}, \Delta S_{m}, S_{d}, R_{i}, V_{i})= F_{1}(\Delta S_{o}) + F_{2}(\Delta S_{m}) + F_{3}(S_{d}) + F_{4}(R_{i}) + F_{5}(V_{i}) + C \end{equation} \begin{equation} \label{f-shapes} F= \sigma S_{o} + \left(\sigma - \gamma_{m} \right) S_{m} + \gamma_{d} S_{d} + \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} + P V_{i} + D \end{equation}

full derivations

note: there may be some residual messyness since I caught a mistake in the sign of \(\gamma_{d}\) in droplet-contact-angle and corrected it after writing all these equations

  • \(F_{1}(\Delta S_{o})\)
    \begin{equation} F_{6}(S_{o}) = \sigma S_{o} \end{equation}

    substituting outside-membrane-area:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-A_{d}^{2}\right) \end{equation}

    substituting droplet-plane-radius:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-(R_{d} \sin \theta_{d})^{2}\right) \end{equation}

    substituting di-cap-radius:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}\right) \end{equation}

    substituting droplet-contact-angle:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 V_{b}}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}

    substituting di-volume:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 (V_{d}+V_{i})}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}

    substituting invagination-volume:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}

    substituting invagination-radius:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}

    "simplified" in mathematica:

    \begin{equation} F_{6}(S_{o}) = \sigma \pi \left(A_t^2+\frac{\left(\gamma _m^2-\gamma _d^2\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(\frac{\gamma _m}{\gamma _d}+2\right) \left(\gamma _d-\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d^2}\right) \end{equation}

    rearranged to isolate constants vs. functions of \(H_{i}\):

    endocytosis phase separation V2 - Page 5.png
    \begin{equation} F_{6}(S_{o}) = \sigma S_{o} \end{equation} \begin{equation} S_{o} = \Delta S_{o} + \pi A_{t}^{2} \end{equation} \begin{equation} F_{6}(S_{o}) = F_{1}(\Delta S_{o}) + \sigma \pi A_{t}^{2} \end{equation} \begin{equation} F_{1}(\Delta S_{o}) = \sigma \Delta S_{o} \end{equation} \begin{equation} \Delta S_{o} = S_{o} - \pi A_{t}^{2} \end{equation} \begin{equation} \Delta S_{o}=\left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi H_{i}^{3}+\frac{3}{2} S_{i} H_{i}+3 V_{d}\right)^{2 / 3} \end{equation}

    replacing invagination height from rel-h:

    \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \Delta S_{o}=\left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{3}+\frac{3}{2} S_{i} h \left(\frac{S_{i}}{\pi}\right)^{1/2}+3 V_{d}\right)^{2 / 3} \end{equation}

    simplified:

    \begin{equation} \Delta S_{o}=\left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \Delta S_{o}=\alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \alpha = \left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \end{equation}
  • \(F_{2}(\Delta S_{m})\)

    gathering from free-energy:

    \begin{equation} \label{} F_{7}(S_{m}) = \left(\sigma - \gamma_{m} \right) S_{m} \end{equation}

    from md-area:

    \begin{equation} \label{md-area} S_{m}=\pi\left(A_{d}^{2}-A_{i}^{2}\right) \end{equation}

    substitute droplet-plane-radius:

    \begin{equation} A_{d}=R_{d} \sin \theta_{d} \end{equation}

    and invagination-plane-radius:

    \begin{equation} A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}} \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(R_{d} \sin \theta_{d}\right)^{2}-\frac{S_{i}}{\pi}+H_{i}^{2}\right) \end{equation}

    rearranged:

    \begin{equation} S_{m}=\pi\left(\left(R_{d} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    substitute di-cap-radius:

    \begin{equation} R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    substitute di-volume:

    \begin{equation} V_{b}=V_{d}+V_{i} \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+V_{i} \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    substitute invagination-volume:

    \begin{equation} V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    substitute invagination-radius:

    \begin{equation} R_{i} =\frac{S_{i}}{2 \pi H_{i}} \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right) \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    substitute droplet-contact-angle:

    \begin{equation} \begin{array}{l} \cos \theta_{d}=\frac{\gamma_{m}}{\gamma_{d}} \\ \sin \theta_{d}=\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}} \end{array} \end{equation}

    to get:

    \begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right) \right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}

    simplified in mathematica:

    \begin{equation} S_{m}=\pi \left(\frac{\left(\gamma _d^2-\gamma _m^2\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(2-\frac{\gamma _m}{\gamma _d}\right) \left(\gamma _d+\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d^2}+H_i^2\right)-S_i \end{equation}

    manually rearranged:

    endocytosis phase separation V2 - Page 6.png
    \begin{equation} \label{} F_{7}(S_{m}) = \left(\sigma - \gamma_{m} \right) S_{m} \end{equation} \begin{equation} S_{m}= \Delta S_{m} - S_{i} \end{equation} \begin{equation} \label{} F_{7}(S_{m}) = F_{2}(\Delta S_{m}) - \left(\sigma - \gamma_{m} \right) S_{i} \end{equation} \begin{equation} \label{} F_{2}(\Delta S_{m}) = \left(\sigma - \gamma_{m} \right) \Delta S_{m} \end{equation} \begin{equation} \Delta S_{m}=\left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi H_{i}^{3}+\frac{3}{2} S_{i} H_{i}+3 V_{d}\right)^{2 / 3}+\pi H_{i}^{2} \end{equation}

    replacing invagination height from rel-h:

    \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \Delta S_{m}=\left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{3}+\frac{3}{2} S_{i} \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)+3 V_{d}\right)^{2 / 3}+ \pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{2} \end{equation}

    simplified:

    \begin{equation} \Delta S_{m}=\left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \Delta S_{m}= \beta \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \beta = \left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \end{equation}
  • \(F_{3}(S_{d})\)

    gathering from free-energy:

    \begin{equation} \label{} F_{3}(S_{d}) = \gamma_{d} S_{d} \end{equation}

    from dc-contact-area:

    \begin{equation} S_{d}=2 \pi R_{d}^{2}\left(1-\cos \theta_{d}\right) \end{equation}

    substitute di-cap-radius

    \begin{equation} R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \end{equation}

    to get:

    \begin{equation} S_{d}=2 \pi \left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}

    substitute di-volume

    \begin{equation} V_{b}=V_{d}+V_{i} \end{equation}

    to get:

    \begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+V_{i}\right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}

    substitute invagination-volume

    \begin{equation} V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{equation}

    to get:

    \begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}

    substitute droplet-contact-angle

    \begin{equation} \cos \theta_{d}=\frac{\gamma_{m}}{\gamma_{d}} \end{equation}

    to get:

    \begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{2 / 3}\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right) \end{equation}

    substitute invagination-radius

    \begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}

    to get:

    \begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{2 / 3}\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right) \end{equation}

    simplified in mathematica:

    \begin{equation} S_{d}= \frac{2 \pi \left(\gamma _d-\gamma _m\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(\frac{\gamma _m}{\gamma _d}+2\right) \left(\gamma _d-\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d} \end{equation}

    manually rearranged:

    20210104_sd.png
    \begin{equation} S_{d}=\left(\frac{8 \pi}{\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)}\right)^{1 / 3}\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} S_{d}=\epsilon \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \epsilon = \left(\frac{8 \pi}{\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}\left(1-\frac{\gamma_{d}}{\gamma_{m}}\right)}\right)^{1 / 3} \end{equation}
  • \(F_{4}(R_{i})\)

    gathering from f-forces:

    \begin{equation} \label{} \begin{array}{l} F_{4}(R_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} \end{array} \end{equation}

    from invagination-radius

    \begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}

    replacing invagination height from rel-h:

    \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} R_{i} = \frac{S_{i}}{2 \pi h \left(\frac{S_{i}}{\pi}\right)^{1/2}} \end{equation}

    simplified:

    \begin{equation} R_{i} = \frac{\sqrt{S_i}}{2 \sqrt{\pi } h} \end{equation}
    • \(F_{4}(h)\)

      substitute invagination-radius

      \begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}

      to get:

      \begin{equation} \label{} \begin{array}{l} F_{4}(H_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{\frac{S_{i}}{2 \pi H_{i}}} - 2 C_{i}\right)^{2} \end{array} \end{equation}

      simplified:

      \begin{equation} \label{} F_{4}(H_{i}) = \frac{2 K}{S_i} \left(C_i S_i-2 \pi H_i\right)^{2} \end{equation}

      replacing invagination height from rel-h:

      \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \label{} F_{4}(h) = \frac{2 K}{S_i} \left(C_i S_i-2 \pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)\right)^{2} \end{equation}

      simplified:

      \begin{equation} \label{} F_{4}(h) = 2 K \left(C_i \sqrt{S_i}-2 \sqrt{\pi } h\right)^2 \end{equation}
  • \(F_{5}(V_{i})\)

    gathering from f-forces:

    \begin{equation} \label{} F_{5}(V_{i}) = P V_{i} \end{equation}

    from invagination-volume-a-h:

    \begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(3 A_{i}^{2}+H_{i}^{2}\right) \end{array} \end{equation}

    substitute invagination-plane-radius

    \begin{equation} \label{} \begin{array}{l} A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}} \end{array} \end{equation}

    to get:

    \begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(3 \left(\frac{S_{i}}{\pi}-H_{i}^{2}\right)+H_{i}^{2}\right) \end{array} \end{equation}

    simplified:

    \begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(\frac{3 S_i}{\pi }-2 H_{i}^2\right) \end{array} \end{equation}

    replacing invagination height from rel-h:

    \begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \label{} V_{i}=\frac{1}{6} \pi h \left(\frac{S_{i}}{\pi}\right)^{1/2}\left(\frac{3 S_i}{\pi }-2 \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^2\right) \end{equation}

    simplified:

    \begin{equation} \label{} V_{i}= \frac{h \left(3-2 h^2\right) S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} \begin{equation} \label{} V_{i}=\eta \left(3 h-2 h^{3}\right) \end{equation} \begin{equation} \label{} \eta=\frac{S_i^{3/2}}{6 \sqrt{\pi }} \end{equation}
  • \(D\): everything else that doesn't change with \(H_{i}\)
    \begin{equation} \label{D} D = \sigma S_{i} - \gamma_{i} S_{i} \end{equation}
  • \(C\): additional constants grouped out from \(\Delta\) terms
    \begin{equation} \label{} C = D + \sigma \pi A_{t}^{2} - \left(\sigma - \gamma_{m} \right) S_{i} \end{equation} \begin{equation} \label{} D = \sigma S_{i} - \gamma_{i} S_{i} \end{equation}

    simplified:

    \begin{equation} \label{} C = \sigma \pi A_t^2+S_i \left(\gamma _m-\gamma _i\right) \end{equation}

DONE substituted after isolating \(h\)

  • could anything be factored out more first?
\begin{equation} \label{} F(h) = \sigma \alpha \xi + \left(\sigma - \gamma_{m} \right) \left( \beta \xi + S_{i} h^{2}\right) + \gamma_{d} \epsilon \xi + 2 K \left(C_i \sqrt{S_i}-2 \sqrt{\pi } h\right)^{2} + P \eta \left(3 h-2 h^{3}\right) + C \end{equation} \begin{equation} \label{} \xi = \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation}
20201203_fh (1).png
\begin{equation} F(h)=\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right)\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3}+\left(\sigma-\gamma_{m}\right) S_{i} h^{2} +2 K\left(C_{i} \sqrt{S}_{i}-2 \sqrt{\pi} h\right)^{2}+P \eta \left(3 h-2 h^{3}\right)+C \end{equation} \begin{equation} F(h)= \left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} - 2 P \eta h^{3}+\left(\left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi\right) h^{2} +\left(3 P \eta-8 C_{i} \sqrt{\pi S_{i}}\right) h+B \end{equation} \begin{equation} B=C+2 K C_{i}^{2} S_{i} \end{equation}

testing sign of alpha terms

import numpy as np
import math
def cube_root(i):
    return math.pow(abs(i),float(1)/3) * (1,-1)[i<0]

def alpha_so(gamma_m, gamma_d):
    alpha = cube_root((np.pi*(gamma_m**2 - gamma_d**2)**3)/((2*gamma_d+gamma_m)**2 * (gamma_d-gamma_m)**4))
    return alpha

for gamma_m in np.linspace(0, 200, 9):
    for gamma_d in np.linspace(0, 200, 9):
        print('gamma_m',gamma_m,'gamma_d',gamma_d,alpha_so(gamma_m, gamma_d))

gamma_m 0.0 gamma_d 0.0 nan
gamma_m 0.0 gamma_d 25.0 -0.9226350743220142
gamma_m 0.0 gamma_d 50.0 -0.9226350743220142
gamma_m 0.0 gamma_d 75.0 -0.9226350743220142
gamma_m 0.0 gamma_d 100.0 -0.9226350743220142
gamma_m 0.0 gamma_d 125.0 -0.9226350743220142
gamma_m 0.0 gamma_d 150.0 -0.9226350743220142
gamma_m 0.0 gamma_d 175.0 -0.9226350743220142
gamma_m 0.0 gamma_d 200.0 -0.9226350743220142
gamma_m 25.0 gamma_d 0.0 1.4645918875615231
gamma_m 25.0 gamma_d 25.0 nan
gamma_m 25.0 gamma_d 50.0 -1.5026501396568157
gamma_m 25.0 gamma_d 75.0 -1.2706753068765375
gamma_m 25.0 gamma_d 100.0 -1.1735039002840684
gamma_m 25.0 gamma_d 125.0 -1.1192302015380546
gamma_m 25.0 gamma_d 150.0 -1.084415611760493
gamma_m 25.0 gamma_d 175.0 -1.0601370721917294
gamma_m 25.0 gamma_d 200.0 -1.042222647325497
gamma_m 50.0 gamma_d 0.0 1.4645918875615231
gamma_m 50.0 gamma_d 25.0 1.7436710272644398
gamma_m 50.0 gamma_d 50.0 nan
gamma_m 50.0 gamma_d 75.0 -1.830739859451904
gamma_m 50.0 gamma_d 100.0 -1.5026501396568157
gamma_m 50.0 gamma_d 125.0 -1.356188576761231
gamma_m 50.0 gamma_d 150.0 -1.2706753068765375
gamma_m 50.0 gamma_d 175.0 -1.214010595446712
gamma_m 50.0 gamma_d 200.0 -1.1735039002840684
gamma_m 75.0 gamma_d 0.0 1.4645918875615231
gamma_m 75.0 gamma_d 25.0 1.590205608287594
gamma_m 75.0 gamma_d 50.0 2.001188208394222
gamma_m 75.0 gamma_d 75.0 nan
gamma_m 75.0 gamma_d 100.0 -2.0727783992021025
gamma_m 75.0 gamma_d 125.0 -1.6820324801561286
gamma_m 75.0 gamma_d 150.0 -1.5026501396568157
gamma_m 75.0 gamma_d 175.0 -1.3955026949998752
gamma_m 75.0 gamma_d 200.0 -1.3231738439535343
gamma_m 100.0 gamma_d 0.0 1.4645918875615231
gamma_m 100.0 gamma_d 25.0 1.5377251238700236
gamma_m 100.0 gamma_d 50.0 1.7436710272644398
gamma_m 100.0 gamma_d 75.0 2.208757298511275
gamma_m 100.0 gamma_d 100.0 nan
gamma_m 100.0 gamma_d 125.0 -2.2692052337015594
gamma_m 100.0 gamma_d 150.0 -1.830739859451904
gamma_m 100.0 gamma_d 175.0 -1.6263744927117951
gamma_m 100.0 gamma_d 200.0 -1.5026501396568157
gamma_m 125.0 gamma_d 0.0 1.4645918875615231
gamma_m 125.0 gamma_d 25.0 1.512803489134373
gamma_m 125.0 gamma_d 50.0 1.6429054603976958
gamma_m 125.0 gamma_d 75.0 1.8801889207945017
gamma_m 125.0 gamma_d 100.0 2.384131644400035
gamma_m 125.0 gamma_d 125.0 nan
gamma_m 125.0 gamma_d 150.0 -2.436744690673985
gamma_m 125.0 gamma_d 175.0 -1.959079850097313
gamma_m 125.0 gamma_d 200.0 -1.7343631139416589
gamma_m 150.0 gamma_d 0.0 1.4645918875615231
gamma_m 150.0 gamma_d 25.0 1.498872430465395
gamma_m 150.0 gamma_d 50.0 1.590205608287594
gamma_m 150.0 gamma_d 75.0 1.7436710272644396
gamma_m 150.0 gamma_d 100.0 2.001188208394222
gamma_m 150.0 gamma_d 125.0 2.5372464543855386
gamma_m 150.0 gamma_d 150.0 nan
gamma_m 150.0 gamma_d 175.0 -2.5840841134673402
gamma_m 150.0 gamma_d 200.0 -2.0727783992021025
gamma_m 175.0 gamma_d 0.0 1.4645918875615234
gamma_m 175.0 gamma_d 25.0 1.4902570606397723
gamma_m 175.0 gamma_d 50.0 1.5585019217103644
gamma_m 175.0 gamma_d 75.0 1.6687875802778238
gamma_m 175.0 gamma_d 100.0 1.836572392913886
gamma_m 175.0 gamma_d 125.0 2.1098678647384412
gamma_m 175.0 gamma_d 150.0 2.6739764366927052
gamma_m 175.0 gamma_d 175.0 nan
gamma_m 175.0 gamma_d 200.0 -2.71637250521393
gamma_m 200.0 gamma_d 0.0 1.4645918875615231
gamma_m 200.0 gamma_d 25.0 1.4845441581729593
gamma_m 200.0 gamma_d 50.0 1.5377251238700236
gamma_m 200.0 gamma_d 75.0 1.6219368867750477
gamma_m 200.0 gamma_d 100.0 1.7436710272644398
gamma_m 200.0 gamma_d 125.0 1.9220789459321215
gamma_m 200.0 gamma_d 150.0 2.208757298511275
gamma_m 200.0 gamma_d 175.0 2.7980755038451366
gamma_m 200.0 gamma_d 200.0 nan
/tmp/ipykernel_172232/638675132.py:7: RuntimeWarning: invalid value encountered in double_scalars
  alpha = cube_root((np.pi*(gamma_m**2 - gamma_d**2)**3)/((2*gamma_d+gamma_m)**2 * (gamma_d-gamma_m)**4))
/tmp/ipykernel_172232/638675132.py:4: DeprecationWarning: In future, it will be an error for 'np.bool_' scalars to be interpreted as an index
  return math.pow(abs(i),float(1)/3) * (1,-1)[i<0]

implementation and analysis [36/42]

analytical derivations

thermodynamic spontaneity (global stability) [5/5]

  • I'm defining "thermodynamic spontaneity" as whether F(h=0) > F(h=1)
  • analyzed by plotting the phase boundary line across 2 parameters when F(h=0) == F(h=1)
  • DONE setting up equality
    20210104_thermodynamic_spontaneity_kfix.png
    \begin{equation} \left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \in\right)\left(3 V_{d}\right)^{2 / 3}+B=\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \in\right)\left(3 V_d+\frac{S_i^{3/2}}{2 \sqrt{\pi }}\right)^{2/3} + \left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi+ P \eta-8 K C_{i} \sqrt{\pi S_{i}}+B \end{equation}
  • DONE solving for \(K\)
    20210104_thermodynamic_k.png
    \begin{equation} \label{therm_spont_k} K = \frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \in\right)\left(\left(3 V_{d}\right)^{2 / 3}-\left(\frac{S_{i}^{3 / 2}}{2 \sqrt{\pi}}+3 V_{d}\right)^{2 / 3}\right)-\left(\sigma-\gamma_{m}\right) S_{i}- P \eta}{8 \pi-8 C_{i} \sqrt{\pi S_{i}}} \end{equation}
  • DONE solving for \(P\)
    20210104_thermodynamic_p.png
    \begin{equation} \label{therm_spont_p} P=\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \in\right)\left(\left(3 V_{d}\right)^{2 / 3}-\left(\frac{S_{i}^{3 / 2}}{2 \sqrt{\pi}}+3 V_{d}\right)^{2 / 3}\right)-\left(\sigma-\gamma_{m}\right) S_{i}-8 K \left(\pi - C_{i} \sqrt{\pi S_{i}}\right)}{\eta} \end{equation}
  • DONE solving for \(\sigma\)
    20210104_thermodynamic_sigma.png
    \begin{equation} \label{therm_spont_sigma} \sigma=\frac{\frac{-\gamma_{m} S_{i}+8 K \left(\pi - C_{i} \sqrt{\pi S_{i}}\right) + P \eta}{\left(3 V_{d}\right)^{2 / 3}-\left(\frac{S_{i}^{3 / 2}}{2 \sqrt{\pi}}+3 V_{d}\right)^{2 / 3}}-\gamma_{d} \epsilon+\gamma_{m} \beta}{\alpha+\beta-\frac{S_{i}}{\left(3 V_{d}\right)^{2 / 3}-\left(\frac{S_{i}^{3 / 2}}{2 \sqrt{\pi}}+3 V_{d}\right)^{2 / 3}}} \end{equation}
  • DONE solving for \(C_{i}\)
    20210104_thermodynamic_ci.png
    \begin{equation} \label{therm_spont_ci} C_{i} = \frac{\pi-\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \in\right)\left(\left(3 V_{d}\right)^{2 / 3}-\left(\frac{S_{i}^{3 / 2}}{2 \sqrt{\pi}}+3 V_{d}\right)^{2 / 3}\right)-\left(\sigma-\gamma_{m}\right) S_{i}-P_{\eta}}{8 K}}{\sqrt{\pi S_{i}}} \end{equation}
  • any others?

kinetic spontaneity (local stability) [11/11]

  • I'm defining "kinetic spontaneity" as whether F'(h=0) is negative or positive
  • DONE solving derivative
    20210105_f_derivative.png
    \begin{equation} \label{f-prime} F^{\prime}(h)=\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right)\left(1-2 h^{2}\right) S_{i}^{3 / 2}}{\sqrt{\pi}\left(\frac{\left(\frac{3 h}{2}-h^{3}\right) S_{i}^{3 / 2}}{\sqrt{\pi}}+3 V_{d}\right)^{1 / 3}}-6 P \eta h^{2}+2\left(\left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi\right) h+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation}
  • DONE solving for h=0
    \begin{equation} \label{f-prime-alpha-beta} F^{\prime}(0)=\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) S_{i}^{3 / 2}}{\sqrt{\pi} \left(3 V_{d}\right)^{1 / 3}}+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation} \begin{equation} \label{f-prime} F^{\prime}(0)=\frac{\left(\gamma_{m} \alpha + \gamma_{d} \epsilon \right) S_{i}^{3 / 2}}{\sqrt{\pi} \left(3 V_{d}\right)^{1 / 3}}+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation}
  • DONE sign of droplet prefactor at h=0
    20210217_prefactor_sign.png

    the sign of the term concerning the droplet must always be positive.

  • DONE solving for \(K\)
    20210106_kinetic_k.png
    \begin{equation} \label{kinetic_spont_k} K=\frac{\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) S_{i}^{3 / 2}}{\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}+3 P \eta}{8 C_{i} \sqrt{\pi S_{i}}} \end{equation}
  • DONE solving for \(P\)
    20210106_kinetic_p.png
    \begin{equation} \label{kinetic_spont_p} P=\frac{\frac{-\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) S_{i}^{3 / 2}}{\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}+8 K C_{i} \sqrt{\pi S_{i}}}{3 \eta} \end{equation}
  • DONE solving for \(\sigma\)
    20210106_kinetic_sigma.png
    \begin{equation} \label{kinetic_spont_sigma} \sigma=\frac{\frac{\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}{S_{i}^{3 / 2}}\left(8 K C_{i} \sqrt{\pi S_{i}}-3 P \eta \right)-\gamma_{d} \epsilon+\gamma_{m} \beta}{(\alpha+\beta)} \end{equation}
  • DONE solving for \(C_{i}\)
    20210106_derivs_kinetic_ci.png
    \begin{equation} \label{kinetic_spont_ci} C_{i} = \frac{\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) S_{i}^{3 / 2}} {\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}+3 P \eta}{8 K \sqrt{\pi S_{i}}} \end{equation}
  • CANCELLED solving for \(\gamma_{m}\)
    20210106_derivs_kinetic_gammam.png
    \begin{equation} \label{kinetic_spont_gammam} \gamma_{m}=\frac{\frac{-\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}{S_{i}^{3 / 2}}\left(8 K C_{i} \sqrt{\pi S_{i}}-3 P \eta \right)+\gamma_{d} \epsilon+\sigma \alpha+\sigma \beta}{\beta} \end{equation}
  • CANCELLED solving for \(\gamma_{d}\)
    20210106_derivs_kinetic_gammad.png
    \begin{equation} \label{kinetic_spont_gammad} \gamma_{d}=\frac{\frac{\sqrt{\pi}\left(3 V_{d}\right)^{1 / 3}}{S_{i}^{3 / 2}}\left(8 K C_{i} \sqrt{\pi S_{i}}-3 P \eta \right)-\sigma \alpha-\left(\sigma-\gamma_{m}\right) \beta}{\epsilon} \end{equation}
  • CANCELLED solving for \(S_{i}\)
  • DONE solving for \(V_{d}\)
    20210106_derivs_kinetic_vd.png
    \begin{equation} \label{kinetic_spont_vd} V_{d}=\frac{\left(\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon \right) S_{i}^{3 / 2}}{\left(8 K C_{i} \sqrt{\pi S_{i}}-3 P \eta \right) \sqrt{\pi}}\right)^{3}}{3} \end{equation}

python code [20/26]

configuration [12/13]

  • import libraries, settings
    %gui qt
import napari
from decimal import Decimal
import math
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt  # plotting
import mpl_toolkits.mplot3d.axes3d as axes3d
%matplotlib inline
import seaborn as sns
import scipy
#sns.set(style='ticks')
#sns.set_palette(sns.cubehelix_palette(10, start=.1, rot=-.75, reverse=True))
sns.set_palette('hls',8)
#sns.set_palette(sns.diverging_palette(250, 30, l=65, center="dark"))
# sns.set_palette("coolwarm", 6)
# from cycler import cycler
SMALL_SIZE = 12
MEDIUM_SIZE = 16
BIGGER_SIZE = 20


plt.rc('axes', titlesize=MEDIUM_SIZE)     # fontsize of the axes title
plt.rc('axes', labelsize=MEDIUM_SIZE)    # fontsize of the x and y labels
plt.rc('xtick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
plt.rc('ytick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
plt.rc('legend', fontsize=SMALL_SIZE)    # legend fontsize
plt.rc('figure', titlesize=BIGGER_SIZE)  # fontsize of the figure title
# default_cycler = (cycler(color=['indigo', 'darkgreen', 'purple', 'green', 'magenta', 'lime']))
# plt.rc('axes', prop_cycle=default_cycler)

pi = np.pi 

#if kernel was started in notebook directory
#fig_dir = 'figures/'
#assuming kernel was started in home directory
fig_dir = 'google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/figures/'

    import IPython
from tabulate import tabulate

class OrgFormatter(IPython.core.formatters.BaseFormatter):
    def __call__(self, obj):
        try:
            return tabulate(obj, headers='keys',
                            tablefmt='orgtbl', showindex='always')
        except:
            return None

ip = get_ipython()
ip.display_formatter.formatters['text/org'] = OrgFormatter()
  • set default fixed parameter values
    • assigned here
      vesicle_radius = 50. #nm
vesicle_diameter = 2*vesicle_radius #nm
drop_radius = 1500. #nm
S_i = 4.*pi*vesicle_radius**2 #nm^2
#H_i_range = np.linspace(0.001*vesicle_radius,2*vesicle_radius,1000) #nm
h_range = np.linspace(0.001,1,1000) #dimensionless 0-1
H_i_range = h_range*(S_i/pi)**(1/2)
sigma = 150. #arbitrary, N/nm^2
A_t = 1000 #nm, arbitrary
K = 100. #arbitrary, N/nm
C_i = 1/vesicle_radius #nm^-1
gamma_d = 200. #arbitrary N/nm^2
gamma_m = 175. #arbitrary, N/nm^2
gamma_i = gamma_m
V_d = 4.*pi*drop_radius**2 #nm^3
P = 1000/(10**9) #N/nm^3

params = {'vesicle_radius' : vesicle_radius,
    'vesicle_diameter' : vesicle_diameter,
    'drop_radius' : drop_radius,
    'S_i' : S_i,
    'h_range' : h_range,
    'H_i_range' : H_i_range,
    'sigma' : sigma,
    'A_t' : A_t,
    'K' : K,
    'C_i' : C_i,
    'gamma_d' : gamma_d,
    'gamma_m' : gamma_m,
    'gamma_i' : gamma_i,
    'V_d' : V_d,
    'P':P}
    • assigned from org table

      established in variables 

      variable_df = pd.DataFrame(variable_list[1:], columns=variable_list[0])
variable_df = variable_df.set_index('python variable')
param_values = variable_df['value']
variable_df
      python variable description value units reference type associated equation LaTeX variable
      H_i invagination height   \(nm\)   input (variable)   \(H_{i}\)
      sigma membrane tension 0.001 \(pN/nm\) (Hassinger et al. 2017) fixed   \(\sigma\)
      A_t total plane radius 2000 \(nm\)   fixed   \(A_{t}\)
      vesicle_radius final vesicle radius 50 \(nm\) (Kaksonen and Roux 2018) fixed   \(R_{v}\)
      K coat bending rigidity 160 \(pN nm\) (Dmitrieff and Nédélec 2015) fixed   \(K\)
      C_i spontaneous curvature 1/50 \(nm^{-1}\) (Hassinger et al. 2017) fixed   \(C_{i}\)
      gamma_d droplet-cytosol surface tension 0.005 \(pN/nm\) (Jawerth et al. 2018) fixed   \(\gamma_{d}\)
      gamma_m droplet-membrane wetting tension 0.002 \(pN/nm\)   fixed   \(\gamma_{m}\)
        initial droplet plane radius       fixed    
      gamma_i droplet-invagination surface tension 0.002 \(pN/nm\)   fixed   \(\gamma_{i}\)
      P turgor pressure 0.1 \(pN/nm^2\) (NO_ITEM_DATA:ma_endocytosis_2019) fixed   \(P\)
      h relative invagination height   unitless   input (variable) rel-h \(h\)
      V_d droplet volume   \(nm^3\)   intermediate output   \(V_{d}\)
      S_i invagination surface area   \(nm^2\)   intermediate output invagination-area \(S_{i}\)
      drop_radius initial droplet spherical radius 600 \(nm\)   intermediate output    
      S_m droplet-flat membrane contact area   \(nm^2\)   intermediate output md-area \(S_{m}\)
      R_i invagination spherical radius   \(nm\)   intermediate output invagination-radius \(R_{i}\)
      S_d droplet-cytosol contact area   \(nm^2\)   intermediate output dc-contact-area \(S_{d}\)
      theta_d droplet contact angle   rad   intermediate output droplet-contact-angle \(\theta_{d}\)
      theta_i invagination contact angle   rad   intermediate output invagination-angle \(\theta_{i}\)
      A_i invagination plane radius   \(nm\)   intermediate output invagination-plane-radius \(A_{i}\)
      A_d droplet + invagination plane radius   \(nm\)   intermediate output droplet-plane-radius \(A_{d}\)
      S_o outside membrane surface area   \(nm^2\)   intermediate output outside-membrane-area \(S_{o}\)
      V_i invagination volume   \(nm^3\)   intermediate output invagination-volume \(V_{i}\)
      V_b droplet + invagination volume   \(nm^3\)   intermediate output di-volume \(V_{b}\)
      H_d droplet + invagination height   \(nm\)   intermediate output di-height \(H_{d}\)
      R_d droplet + invagination spherical radius   \(nm\)   intermediate output di-cap-radius \(R_{d}\)
      F free energy   \(pN \cdot nm\)   output f-forces f-shapes \(F\)
      membrane_tension membrane tension energy   \(pN \cdot nm\)   output    
      bending_energy bending energy   \(pN \cdot nm\)   output    
      surface_tension surface tension energy   \(pN \cdot nm\)   output    
      wetting_energy wetting energy   \(pN \cdot nm\)   output    
      turgor_pressure turgor pressure energy   \(pN \cdot nm\)   output    
      2÷R_i membrane curvature   \(nm^{-1}\)   output     
      vesicle_radius = param_values['vesicle_radius']
vesicle_diameter = 2*vesicle_radius #nm
drop_radius = param_values['drop_radius'] #nm
S_i = 4.*pi*vesicle_radius**2 #nm^2
#H_i_range = np.linspace(0.001*vesicle_radius,2*vesicle_radius,1000) #nm
h_range = np.linspace(0.001,1,1000) #dimensionless 0-1
H_i_range = h_range*(S_i/pi)**(1/2)
sigma = param_values['sigma'] #arbitrary, N/nm^2
A_t = param_values['A_t'] #nm, arbitrary
K = param_values['K'] #arbitrary, N/nm
C_i = 1/vesicle_radius #nm^-1
gamma_d = param_values['gamma_d'] #arbitrary N/nm^2
gamma_m = param_values['gamma_m'] #arbitrary, N/nm^2
gamma_i = gamma_m
V_d = 4.*pi*drop_radius**2 #nm^3
P = param_values['P'] #N/nm^2

params = {'vesicle_radius' : vesicle_radius,
    'vesicle_diameter' : vesicle_diameter,
    'drop_radius' : drop_radius,
    'S_i' : S_i,
    'h_range' : h_range,
    'H_i_range' : H_i_range,
    'sigma' : sigma,
    'A_t' : A_t,
    'K' : K,
    'C_i' : C_i,
    'gamma_d' : gamma_d,
    'gamma_m' : gamma_m,
    'gamma_i' : gamma_i,
    'V_d' : V_d,
    'P':P}
  • define composite variables

    This is just to check what they are, I re-define them whenever I solve for F 

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
gamma_ratio = gamma_m/gamma_d
alpha_refactored = np.cbrt((pi*(gamma_ratio+1)**3)/((gamma_ratio-1)*(gamma_ratio+2)**2))
beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
#epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
epsilon = np.cbrt((8*pi)/((2+gamma_ratio)**2*(1-gamma_ratio)))
eta = (S_i**(3/2))/(6*pi**(1/2))
C = sigma*pi*A_t**2+S_i*(gamma_m-gamma_i)

alpha, alpha_refactored, beta, epsilon, eta, C
    -1.3561885767612312 -1.356188576761231 1.3561885767612312 1.937412252516044 523598.7755982989 12566.370614359173
  • custom functions [12/13]
    • model functions
      • pre-analytical version
        • [ ] rewrite each function variable as dictionary key (outpus['theta_d'] instead of theta_d for the first one) and just return outputs at end

          • should eliminate some redundancy 

             def llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
               C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d,
               P=P):
     #H_i = h*(S_i/pi)**(1/2)
     # might need to be -gamma_m/gamma_d
     theta_d = np.arccos(gamma_m/gamma_d) #equation 8
     #to make a full array out of the constant theta_d value
     #theta_d = np.full_like(H_i, theta_d)  
     A_i = ((S_i/pi)-H_i**2)**(1/2) #equation 7
     V_i = (1/6)*pi*H_i*(3*A_i**2+H_i**2) #equation 6
 # this is actually wrong!
 #    R_i = (A_i**2-H_i**2)/(2*H_i) #equation 10
     R_i = S_i/(2*pi*H_i) #equation 10 re-do
 # this is wrong!
 #    theta_i = np.arcsin(A_i/R_i) #equation 12
     theta_i = np.arccos(1-S_i/(2*pi*R_i**2)) #equation 12 re-do
     V_b = V_d+V_i # equation 5
#      R_d_list = []
#      for V in V_b:
#          R_d_list.append(cube_root((3*V)/(pi*(2+np.cos(theta_d))*(1-np.cos(theta_d))**2))) 
     R_d = ((3*V_b)/(pi*(2+np.cos(theta_d))*(1-np.cos(theta_d))**2))**(1/3) #equation 4
     # R_d = np.array(R_d_list)
     S_d = 2*pi*R_d**2*(1-np.cos(theta_d)) #equation 11
     A_d = R_d*np.sin(theta_d) #equation 3
     S_m = pi*(A_d**2-A_i**2) #equation 9
     S_o = pi*(A_t**2-A_d**2) #equation 2
     membrane_tension = sigma*(S_o+S_m+S_i)
     bending_energy = (K/2)*S_i*((2/R_i)-(2*C_i))**2
     surface_tension = gamma_d*S_d
     wetting_energy = gamma_m*S_m+gamma_i*S_i
     droplet_energy = surface_tension - wetting_energy
     turgor_pressure = P*V_i
     F = membrane_tension+bending_energy+surface_tension-wetting_energy+turgor_pressure #equation 1

     outputs = {'theta_d':theta_d,'A_i':A_i,'V_i':V_i, 'R_i':R_i,
                '2÷R_i':2/R_i,'theta_i':theta_i,'V_b':V_b,'R_d':R_d,
               'S_d':S_d,'A_d':A_d,'S_m':S_m,'S_o':S_o,
                'F':F,'membrane_tension':membrane_tension,
                'bending_energy':bending_energy, 'bending_energy_flipped':-bending_energy,
                'surface_tension':surface_tension,
                'wetting_energy':-wetting_energy, 'wetting_energy_flipped':wetting_energy,
                'droplet_energy':droplet_energy, 'turgor_pressure':turgor_pressure}
     return outputs

        plt.plot(H_i_range/100,llps_model()['F'])
plt.xlabel('h')
plt.ylabel('F')
plt.tight_layout()
plt.savefig(fig_dir+'llps_model_test.png')

        ---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
Cell In [9], line 5
      3 plt.ylabel('F')
      4 plt.tight_layout()
----> 5 plt.savefig(fig_dir+'llps_model_test.png')

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/pyplot.py:977, in savefig(*args, **kwargs)
    974 @_copy_docstring_and_deprecators(Figure.savefig)
    975 def savefig(*args, **kwargs):
    976     fig = gcf()
--> 977     res = fig.savefig(*args, **kwargs)
    978     fig.canvas.draw_idle()   # need this if 'transparent=True' to reset colors
    979     return res

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/figure.py:3058, in Figure.savefig(self, fname, transparent, **kwargs)
   3054     for ax in self.axes:
   3055         stack.enter_context(
   3056             ax.patch._cm_set(facecolor='none', edgecolor='none'))
-> 3058 self.canvas.print_figure(fname, **kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backend_bases.py:2319, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2315 try:
   2316     # _get_renderer may change the figure dpi (as vector formats
   2317     # force the figure dpi to 72), so we need to set it again here.
   2318     with cbook._setattr_cm(self.figure, dpi=dpi):
-> 2319         result = print_method(
   2320             filename,
   2321             facecolor=facecolor,
   2322             edgecolor=edgecolor,
   2323             orientation=orientation,
   2324             bbox_inches_restore=_bbox_inches_restore,
   2325             **kwargs)
   2326 finally:
   2327     if bbox_inches and restore_bbox:

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backend_bases.py:1648, in _check_savefig_extra_args.<locals>.wrapper(*args, **kwargs)
   1640     _api.warn_deprecated(
   1641         '3.3', name=name, removal='3.6',
   1642         message='%(name)s() got unexpected keyword argument "'
   1643                 + arg + '" which is no longer supported as of '
   1644                 '%(since)s and will become an error '
   1645                 '%(removal)s')
   1646     kwargs.pop(arg)
-> 1648 return func(*args, **kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/_api/deprecation.py:415, in delete_parameter.<locals>.wrapper(*inner_args, **inner_kwargs)
    405     deprecation_addendum = (
    406         f"If any parameter follows {name!r}, they should be passed as "
    407         f"keyword, not positionally.")
    408     warn_deprecated(
    409         since,
    410         name=repr(name),
   (...)
    413                  else deprecation_addendum,
    414         **kwargs)
--> 415 return func(*inner_args, **inner_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backends/backend_agg.py:541, in FigureCanvasAgg.print_png(self, filename_or_obj, metadata, pil_kwargs, *args)
    494 """
    495 Write the figure to a PNG file.
    496 
   (...)
    538     *metadata*, including the default 'Software' key.
    539 """
    540 FigureCanvasAgg.draw(self)
--> 541 mpl.image.imsave(
    542     filename_or_obj, self.buffer_rgba(), format="png", origin="upper",
    543     dpi=self.figure.dpi, metadata=metadata, pil_kwargs=pil_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/image.py:1675, in imsave(fname, arr, vmin, vmax, cmap, format, origin, dpi, metadata, pil_kwargs)
   1673 pil_kwargs.setdefault("format", format)
   1674 pil_kwargs.setdefault("dpi", (dpi, dpi))
-> 1675 image.save(fname, **pil_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/PIL/Image.py:2317, in Image.save(self, fp, format, **params)
   2315         fp = builtins.open(filename, "r+b")
   2316     else:
-> 2317         fp = builtins.open(filename, "w+b")
   2319 try:
   2320     save_handler(self, fp, filename)

FileNotFoundError: [Errno 2] No such file or directory: 'google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/figures/llps_model_test.png'
        c8fe3356fecff67fa663cdfd814c256ec1d29d36.png
      • post-analytical version
        def llps_model_recbrt(h=h_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
               C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i,
               V_d=V_d, P=P):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))
    C = sigma*pi*A_t**2+S_i*(gamma_m-gamma_i)

    F = ((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)* np.cbrt(((S_i**(3/2))/(pi**(1/2)))*(3*h/2-h**3)+3*V_d)**2+ (sigma-gamma_m)*S_i*h**2+ 2*K*(C_i*S_i**(1/2)-2*pi**(1/2)*h)**2+ P*eta*(3*h-2*h**3)+ C )

#    print(alpha, beta, epsilon, eta, C)

    return F

        plt.plot(h_range,llps_model_recbrt())
plt.xlabel('h')
plt.ylabel('F')
plt.tight_layout()
plt.savefig(fig_dir+'llps_model_recbrt_test.png')

        ---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
Cell In [11], line 5
      3 plt.ylabel('F')
      4 plt.tight_layout()
----> 5 plt.savefig(fig_dir+'llps_model_recbrt_test.png')

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/pyplot.py:977, in savefig(*args, **kwargs)
    974 @_copy_docstring_and_deprecators(Figure.savefig)
    975 def savefig(*args, **kwargs):
    976     fig = gcf()
--> 977     res = fig.savefig(*args, **kwargs)
    978     fig.canvas.draw_idle()   # need this if 'transparent=True' to reset colors
    979     return res

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/figure.py:3058, in Figure.savefig(self, fname, transparent, **kwargs)
   3054     for ax in self.axes:
   3055         stack.enter_context(
   3056             ax.patch._cm_set(facecolor='none', edgecolor='none'))
-> 3058 self.canvas.print_figure(fname, **kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backend_bases.py:2319, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2315 try:
   2316     # _get_renderer may change the figure dpi (as vector formats
   2317     # force the figure dpi to 72), so we need to set it again here.
   2318     with cbook._setattr_cm(self.figure, dpi=dpi):
-> 2319         result = print_method(
   2320             filename,
   2321             facecolor=facecolor,
   2322             edgecolor=edgecolor,
   2323             orientation=orientation,
   2324             bbox_inches_restore=_bbox_inches_restore,
   2325             **kwargs)
   2326 finally:
   2327     if bbox_inches and restore_bbox:

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backend_bases.py:1648, in _check_savefig_extra_args.<locals>.wrapper(*args, **kwargs)
   1640     _api.warn_deprecated(
   1641         '3.3', name=name, removal='3.6',
   1642         message='%(name)s() got unexpected keyword argument "'
   1643                 + arg + '" which is no longer supported as of '
   1644                 '%(since)s and will become an error '
   1645                 '%(removal)s')
   1646     kwargs.pop(arg)
-> 1648 return func(*args, **kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/_api/deprecation.py:415, in delete_parameter.<locals>.wrapper(*inner_args, **inner_kwargs)
    405     deprecation_addendum = (
    406         f"If any parameter follows {name!r}, they should be passed as "
    407         f"keyword, not positionally.")
    408     warn_deprecated(
    409         since,
    410         name=repr(name),
   (...)
    413                  else deprecation_addendum,
    414         **kwargs)
--> 415 return func(*inner_args, **inner_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/backends/backend_agg.py:541, in FigureCanvasAgg.print_png(self, filename_or_obj, metadata, pil_kwargs, *args)
    494 """
    495 Write the figure to a PNG file.
    496 
   (...)
    538     *metadata*, including the default 'Software' key.
    539 """
    540 FigureCanvasAgg.draw(self)
--> 541 mpl.image.imsave(
    542     filename_or_obj, self.buffer_rgba(), format="png", origin="upper",
    543     dpi=self.figure.dpi, metadata=metadata, pil_kwargs=pil_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/image.py:1675, in imsave(fname, arr, vmin, vmax, cmap, format, origin, dpi, metadata, pil_kwargs)
   1673 pil_kwargs.setdefault("format", format)
   1674 pil_kwargs.setdefault("dpi", (dpi, dpi))
-> 1675 image.save(fname, **pil_kwargs)

File ~/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/PIL/Image.py:2317, in Image.save(self, fp, format, **params)
   2315         fp = builtins.open(filename, "r+b")
   2316     else:
-> 2317         fp = builtins.open(filename, "w+b")
   2319 try:
   2320     save_handler(self, fp, filename)

FileNotFoundError: [Errno 2] No such file or directory: 'google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/figures/llps_model_recbrt_test.png'
        c8fe3356fecff67fa663cdfd814c256ec1d29d36.png
    • general plotting function
      • figure out some way to get the right units for each thing being plotted
      def plot_simulation(x=H_i_range/vesicle_diameter, output='F',
                 H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                 C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                 label=None, variable_df=variable_df):
    outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
              C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)
    y = outputs[output]
    plt.plot(x,y,label=label)
    plt.ylabel(variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')')
      • testing
        plot_simulation()
        07513f6365b6a9eb35be95f9bf0c21ac689164c3.png
    • normalized plots
      def plot_simulation_normalized(x=H_i_range/vesicle_diameter, output='F',
                 H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                 C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                 label=None):
    outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
              C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d)
    y_sub = outputs[output]-np.min(outputs[output])
    y_norm = y_sub/np.max(y_sub)
    plt.plot(x,y_norm,label=label)
    plt.ylabel(output)
    plt.xlabel('invagination distance/vesicle diameter')
    • delta plots
      def plot_simulation_delta(x=H_i_range/vesicle_diameter, output='F',
                 H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                 C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                 label=None, variable_df=variable_df, color=None):
    outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
              C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)
    output_0 = llps_model(H_i=H_i_range[0], sigma=sigma, A_t=A_t, S_i=S_i, K=K,
              C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)
    y = outputs[output]-output_0[output]
    plt.plot(x,y,label=label, color=color)
    #plt.ylabel('∆ '+variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')')
    plt.xlabel('h: invagination distance/vesicle diameter')
    • show geometry for given invagination distance
      def show_geometry(H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                  C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                  fig_width=10, fig_height=5,
                  num_plots=1, top=20, bottom=-250, left=-500, right=500):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
             C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    R_d = outputs['R_d']
    R_i = outputs['R_i']

    #fig, ax = plt.subplots(1,num_plots)
    #plt.subplot(1, num_plots, 1)

    fig = plt.figure()
    fig.set_size_inches((fig_width,fig_height))
    ax = fig.add_subplot(1, num_plots, 1)

    plt.hlines([0,0],[-A_t,A_d],[-A_d,A_t], color='r', label='$S_o$')
    plt.hlines([0,0],[-A_d,A_i],[-A_i,A_d], color='g', label='$S_m$')

    if V_d > 0:
      points_d = np.linspace(pi/2-theta_d, theta_d+pi/2, 1000)
      x_d = R_d*np.cos(points_d)
      y_d = R_d*np.sin(points_d)
      y_d_adj = y_d-np.min(y_d)
      plt.plot(x_d,-y_d_adj, color='b', label='$S_d$')

    points_i = np.linspace(pi/2-theta_i, theta_i+pi/2, 1000)
    x_i = R_i*np.cos(points_i)
    y_i = R_i*np.sin(points_i)
    y_i_adj = y_i-np.min(y_i)
    plt.plot(x_i,-y_i_adj, color='purple', label='$S_i$')

    ax.set_ylim(top=top, bottom=bottom)
    ax.set_xlim(left=left, right=right)
    ax.set_aspect('equal')
    ax.set(xlabel = 'distance along membrane (nm)', 
           ylabel = 'distance into cytoplasm (nm)')
    plt.legend()
    plt.tight_layout()

    if num_plots > 1:
      return fig
      • testing
        show_geometry(H_i=100)
        dc1240c7e73de4aaa6b4c168b30fcd0ff3844e07.png
    • 3D geometry
      • points
        %gui qt
viewer = napari.Viewer(ndisplay=3)

spherecoords = []
for H_i in np.linspace(1,100,100):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_d, np.pi, 200)
    THETA, PHI = np.meshgrid(theta, phi)
    R = R_d
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)
    height = np.full_like(Z, H_i)
    spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T)

spherecoords = np.concatenate(spherecoords)
viewer.add_points(np.array(spherecoords), name='droplet', size=20, face_color='cyan',
                  edge_color = 'transparent', blending='additive', opacity=0.6)

spherecoords = []
for H_i in np.linspace(1,100,100):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_i, np.pi, 200)
    THETA, PHI = np.meshgrid(theta, phi)
    R = R_i
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)
    height = np.full_like(Z, H_i)
    spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T)

spherecoords = np.concatenate(spherecoords)
viewer.add_points(np.array(spherecoords), name='coat', size=20, face_color='purple',
                  edge_color = 'transparent', blending='additive', opacity=0.6)

spherecoords = []
for H_i in np.linspace(1,100,100):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    theta_m = 0.00000001
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_m, np.pi, 200)
    THETA, PHI = np.meshgrid(theta, phi)
    R = 0.1*A_t/np.sin(theta_m)
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)+0
    height = np.full_like(Z, H_i)
    spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T)

spherecoords = np.concatenate(spherecoords)
viewer.add_points(np.array(spherecoords), name='cell wall', size=20, face_color='red',
                  edge_color = 'transparent', blending='additive', opacity=0.6)


        <Points layer 'cell wall' at 0x7f3cb42b7ca0>
        • I don't understand why this doesn't really work
        vertices = spherecoords
#faces = np.array([[0, 0, 1, 2], [0, 1, 2, 3], [0, 2, 3, 4]])
faces_list = []
counter = 0
for H_i in range(len(H_i_range)):
    for point in range(38):
        pointa = counter + point
        faces_list.append([pointa, pointa+1, pointa+2])
    counter += 40
faces = np.array(faces_list)
values = np.linspace(0, 1, len(vertices))
surface = (vertices, faces, values)

viewer.add_surface(surface, colormap='viridis')  # add the surface

        <Surface layer 'surface' at 0x7f9405b20130>
      • surface

        examples: 

        vertices = np.array([[0, 0, 0, 0], [0, 10, 0, 20], [0, 20, 10, 0], [0, 0, 10, 10],
                     [1, 20, 0, 0], [1, 10, 0, 20], [1, 20, 10, 0], [1, 0, 10, 10]])
faces = np.array([[0, 1, 2], [1, 2, 3],
                  [4,5,6],[5,6,7]])
values = np.linspace(0.9, 1, len(vertices))
surface = (vertices, faces, values)

viewer = napari.view_surface(surface)  # add the surface

facesloop = []

for t in (0,1):
    vert = vertices[vertices[:,0] == t]
    facest = scipy.spatial.Delaunay(vert[:,2:]).simplices + len(vert)*t
    facesloop.append(facest)

facesd = np.concatenate(facesloop)

viewer = napari.view_surface((vertices, facesd, values))

        coords = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T[:,1:]
faces = scipy.spatial.Delaunay( coords[:,1:]).simplices
values = np.linspace(0,1,len(coords))
viewer = napari.view_surface((coords[:,1:], faces, values))

        %gui qt
viewer = napari.Viewer(ndisplay=3)

spherecoords = []
faces = []
for H_i in range(1,101):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_d, np.pi, 50)
    THETA, PHI = np.meshgrid(theta, phi)
    R = R_d
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)
    height = np.full_like(Z, H_i)
    vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T
    spherecoords.append(vertices)

    #face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1)
    if H_i == 1:
        face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1)
    else:
        face = faces[-1] + len(vertices)
    faces.append(face)

spherecoords = np.concatenate(spherecoords)
faces = np.concatenate(faces)
surface_values = np.full(len(spherecoords),1)
viewer.add_surface((spherecoords,faces,surface_values), name='droplet', colormap='green',
                   blending='translucent', shading='smooth')

spherecoords = []
faces = []
for H_i in range(1,101):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_i, np.pi, 50)
    THETA, PHI = np.meshgrid(theta, phi)
    R = R_i
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)
    height = np.full_like(Z, H_i)
    vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T
    spherecoords.append(vertices)

    if H_i == 1:
        face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1)
    else:
        face = faces[-1] + len(vertices)
    faces.append(face)

spherecoords = np.concatenate(spherecoords)
faces = np.concatenate(faces)
surface_values = np.full(len(spherecoords),1)
viewer.add_surface((spherecoords,faces,surface_values), name='coat', colormap='bop blue',
                   blending='additive', shading='smooth')

spherecoords = []
faces = []
for H_i in range(1,101):
    outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

    A_d = outputs['A_d']
    A_i = outputs['A_i']
    theta_d = outputs['theta_d']
    theta_i = outputs['theta_i']
    theta_m = 0.00000001
    R_d = outputs['R_d']
    R_i = outputs['R_i']
    theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_m, np.pi, 50)
    THETA, PHI = np.meshgrid(theta, phi)
    R = 0.1*A_t/np.sin(theta_m)
    X = R * np.sin(PHI) * np.cos(THETA)
    Y = R * np.sin(PHI) * np.sin(THETA)
    Z = R * np.cos(PHI)
    Z = Z-np.max(Z)+0
    height = np.full_like(Z, H_i)
    vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T
    spherecoords.append(vertices)

    #face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1)
    if H_i == 1:
        face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1)
    else:
        face = faces[-1] + len(vertices)
    faces.append(face)

spherecoords = np.concatenate(spherecoords)
faces = np.concatenate(faces)
surface_values = np.full(len(spherecoords),1)
viewer.add_surface((spherecoords,faces,surface_values), name='cell wall', colormap='bop blue',
                   blending='opaque', shading='smooth')

def update_frame():
    """Update frame or time"""
    viewer.text_overlay.text = 'h = %1.2f\nblue: membrane\ngreen: condensate' % ((viewer.dims.current_step[0]+1)/100)

viewer.dims.events.current_step.connect(lambda event: update_frame())

viewer.text_overlay.visible=True
viewer.text_overlay.font_size=20
viewer.text_overlay.color = 'white'
viewer.scale_bar.unit = 'nm'
viewer.scale_bar.visible=True
viewer.scale_bar.font_size=10

        animation script 

        from napari_animation import Animation

animation = Animation(viewer)
viewer.reset_view()
viewer.dims.current_step = (0, 0,0,0)
animation.capture_keyframe(steps=15)
animation.capture_keyframe(steps=15)
viewer.camera.zoom = 2.6
viewer.camera.angles=(-0.026467524771271397, 0.5719902891444765, 11.550928988495166)
animation.capture_keyframe(steps=50)
viewer.dims.current_step = (viewer.dims.range[0][1]-1, 0,0,0)
animation.capture_keyframe(steps=int(viewer.dims.range[0][1]-1))
viewer.reset_view()
animation.capture_keyframe(steps=50)
viewer.dims.current_step = (0, 0,0,0)
animation.capture_keyframe(steps=int(viewer.dims.range[0][1]-1))
animation.animate('~/google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/scripted_surface_animation.mov')

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      • probably don't need this stuff
        num_plots = 1
outputs = llps_model(H_i=90, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P)

A_d = outputs['A_d']
A_i = outputs['A_i']
theta_d = outputs['theta_d']
theta_i = outputs['theta_i']
R_d = outputs['R_d']
R_i = outputs['R_i']

#fig, ax = plt.subplots(1,num_plots)
#plt.subplot(1, num_plots, 1)

#fig = plt.figure()
#fig.set_size_inches((fig_width,fig_height))
#ax = fig.add_subplot(1, num_plots, 1)

#plt.hlines([0,0],[-A_t,A_d],[-A_d,A_t], color='r', label='$S_o$')
#plt.hlines([0,0],[-A_d,A_i],[-A_i,A_d], color='g', label='$S_m$')

if V_d > 0:
    points_d = np.linspace(pi/2-theta_d, theta_d+pi/2, 1000)
    x_d = R_d*np.cos(points_d)
    y_d = R_d*np.sin(points_d)
    y_d_adj = y_d-np.min(y_d)

points_i = np.linspace(pi/2-theta_i, theta_i+pi/2, 1000)
x_i = R_i*np.cos(points_i)
y_i = R_i*np.sin(points_i)
y_i_adj = y_i-np.min(y_i)

xy_i = np.array([y_i,x_i]).T
viewer = napari.Viewer()
viewer.add_points(xy_i)

        <Points layer 'xy_i' at 0x7f9405887370>

        nx, ny, nz = (3, 3, 4)
x = np.linspace(0, 10, nx)
y = np.linspace(0, 10, ny)
z = np.linspace(0, 10, nz)

xv, yv, zv = np.meshgrid(x, y, z)
zyxv = np.array([np.ravel(zv), np.ravel(yv), np.ravel(xv)]).T
zyxv

viewer = napari.Viewer()
viewer.add_points(zyxv)

        <Points layer 'zyxv' at 0x7f93f2b41fd0>

        import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d

ax = plt.axes(projection="3d")
xlist=np.linspace(-1.0,1.0,10)*100
ylist=np.linspace(-1.0,1.0,10)*100
r=np.linspace(1.0,1.0,10)*100
X,Y= np.meshgrid(xlist,ylist)

Z=np.sqrt(r**2-X**2-Y**2) #Use np.sqrt like you had before

spherecoords = np.array([np.ravel(Z), np.ravel(Y), np.ravel(X)]).T
viewer.add_points(spherecoords)

vertices = spherecoords
faces = np.array([[0, 50, 99], [1, 2, 3]])
values = np.linspace(1, 1, len(vertices))
surface = (vertices, faces, values)

viewer.add_surface(surface)

cp=ax.plot_wireframe(X,Y,Z,color="r")
cp=ax.plot_wireframe(X,Y,-Z,color="r") # Now plot the bottom half


plt.title('2D Contour Plot of The unit sphere')
plt.show()

        <ipython-input-21-c34f0052e1a6>:11: RuntimeWarning: invalid value encountered in sqrt
  Z=np.sqrt(r**2-X**2-Y**2) #Use np.sqrt like you had before
        e0cc5e561ea03e27b011a913d05aff6ff077c5fd.png
    • side-by-side delta plot and equilibrium geometry
      def plot_simulation_geometry(x=H_i_range/vesicle_diameter, output='F',
                 H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                 C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                 fig_width=15, fig_height=5, label=None, variable_df=variable_df,
                num_plots=2, top=20, bottom=-250, left=-500, right=500):
    outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
              C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d)
    min_energy_ix = np.argmin(outputs['F'])
    H_i_eq = H_i_range[min_energy_ix]
    fig = show_geometry(H_i=H_i_eq, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                   C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                   fig_width=fig_width, fig_height=fig_height,
                   num_plots=num_plots, top=top, bottom=bottom, left=left, right=right)
    plt.title('geometry at equilibrium')
    ax2 = fig.add_subplot(1,2,2)
    #y = outputs[output]
    #plt.plot(x,y,label=label)
    #plt.ylabel(output)

    plot_simulation_delta(x=x, output=output,
                 H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K,
                 C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P,
                 label=label, variable_df=variable_df)

    plt.tight_layout()
      • testing
        plot_simulation_geometry()
        919d3104ddd84a014e93996a89df4a09f5207fd8.png
    • side-by-side F plots and parameter phase diagram
      def boundary_test(x_param, y_param, x_range, boundary, params=params,
                  factors=[0.7,1,1.3], y_offset=0.2, x_scale='log'):
    fparams = params.copy()
    fig, (ax1, ax2)  = plt.subplots(nrows=1, ncols=2, figsize=(10,4))

    if x_scale == 'log':
        ax1.semilogx(x_range, boundary, color='black')
    else:
        ax1.plot(x_range, boundary, color='black')

    for factor in factors:
        middle = len(x_range)//2
        x_pick = x_range[middle]*factor
        boundary_pick = boundary[middle]+(factor-1)*np.sqrt(boundary[middle]**2)*y_offset
        fparams[x_param] = x_pick
        fparams[y_param] = boundary_pick
        F = llps_model_recbrt(h=fparams['h_range'], sigma=fparams['sigma'], A_t=fparams['A_t'],
                S_i=fparams['S_i'], K=fparams['K'], C_i=fparams['C_i'], gamma_d=fparams['gamma_d'],
                gamma_m=fparams['gamma_m'], gamma_i=fparams['gamma_i'], V_d=fparams['V_d'], P=fparams['P'])
        delta_F = F-F[0]

        ax1.scatter(x_pick, boundary_pick)
        ax2.plot(h_range,delta_F)

    ax2.hlines(0, 0,1.05, color='black')

    ax1.set(xlabel=x_param, ylabel=y_param)
    ax2.set(xlabel='h', ylabel='∆F')
    plt.tight_layout()
    • global stability parameter analysis [4/4]
      • DONE \(K\)

        from therm_spont_k 

        def thermo_spont_K(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, P=P, C_i=C_i):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    K = ((-(sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)*
          (np.cbrt(3*V_d+(S_i**(3/2))/(2*pi**(1/2)))**2-
            np.cbrt(3*V_d)**2)+2*P*eta-(sigma-gamma_m)*S_i-
          3*P*eta)/(8*pi-8*C_i*(pi*S_i)**(1/2)))

    return K
        • testing
          V_d_range = np.logspace(6,9)
K_boundary = thermo_spont_K(V_d=V_d_range)
boundary_test(x_param='V_d', y_param='K', x_range=V_d_range, boundary=K_boundary)
          665a7d83f843680a7819f32b40f31c9caf4b42f7.png 
          C_i_range = np.logspace(-4,-2.1,1000)
K_boundary = thermo_spont_K(C_i=C_i_range)
boundary_test(x_param='C_i', y_param='K', x_range=C_i_range, boundary=K_boundary,
              y_offset=-0.3)
          d139b6fe1b14eea2f8fcc8c11e535077007730a2.png
          • this kind of makes sense - the lower the spontaneous curvature (flatter), the more the energetics are sensitive to rigidity
          x_range = np.logspace(6,9)
thermo_boundary = thermo_spont_K(V_d=x_range)
kinetic_boundary = kinetic_spont_K(V_d=x_range)
both_boundary_test(x_param='V_d', y_param='K', x_range=x_range,
                   thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                   y_offset=2)
          e814f99585c220b4dde760e2b4698507a17f12a6.png
      • DONE \(P\)
        def thermo_spont_P(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, K=K, C_i=C_i):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    P = ((sigma*alpha + (sigma - gamma_m)*beta + gamma_d*epsilon)*(np.cbrt(3*V_d)**2 -
         np.cbrt(S_i**(3/2)/(2*np.sqrt(pi)) + 3*V_d)**2) -
         (sigma-gamma_m)*S_i - 8*K*(pi - C_i*np.sqrt(pi*S_i)))/ eta

    return P
        • testing
          x_range = np.logspace(6,9)
boundary = thermo_spont_P(V_d=x_range)
boundary_test(x_param='V_d', y_param='P', x_range=x_range, boundary=boundary,
              y_offset=-0.3)
          dbefcf0df68c98f4f818056fbe6d31aefd1d9a1b.png
      • DONE \(\sigma\)
        def thermo_spont_sigma(P=P, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, K=K, C_i=C_i):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    sigma = ((-gamma_m*S_i+8*K*(pi-C_i*np.sqrt(pi*S_i))+P*eta)/
            ((3*V_d)**(2/3)-((S_i**(3/2))/
                (2*np.sqrt(pi))+3*V_d)**(2/3))-
             gamma_d*epsilon+gamma_m*beta)/(alpha+beta-
            (S_i)/((3*V_d)**(2/3)-((S_i**(3/2))/
                    (2*np.sqrt(pi))+3*V_d)**(2/3)))

    return sigma
        • testing
          x_range = np.logspace(6,9)
boundary = thermo_spont_sigma(V_d=x_range)
boundary_test(x_param='V_d', y_param='sigma', x_range=x_range, boundary=boundary,
              y_offset=-0.02)
          56de5a9bde3d9715633bcb7f298c02800d3679b8.png
      • DONE \(C_{i}\)
        def thermo_spont_C_i(P=P, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, K=K, sigma=sigma):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    C_i = (pi-((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)*(np.cbrt(3*V_d)**2-np.cbrt((S_i**(3/2))/(2*np.sqrt(pi))+3*V_d)**2)-(sigma-gamma_m)*S_i-P*eta)/(8*K))/(np.sqrt(pi*S_i))

    return C_i
        • testing
          x_range = np.logspace(2,7)
boundary = thermo_spont_C_i(V_d=x_range)
boundary_test(x_param='V_d', y_param='C_i', x_range=x_range, boundary=boundary,
              y_offset=0.2)
          dfa200386df3634b6222738d978b1475b912a8df.png
          • it doesn't make sense that C_i would ever be negative…
    • local stability parameter analysis [8/8]
      • DONE \(K\)

        from kinetic_spont_k 

        def kinetic_spont_K(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, P=P, C_i=C_i):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    K = (((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)*S_i**(3/2))/(np.sqrt(pi)*np.cbrt(3*V_d)) + 3*P*eta)/(8*C_i*np.sqrt(pi*S_i))

    return K
        • testing
          x_range = np.logspace(6,9)
boundary = kinetic_spont_K(V_d=x_range)
boundary_test(x_param='V_d', y_param='K', x_range=x_range, boundary=boundary,
              y_offset=0.5)
          e4cf69318c7a310e9ff8a21f7763fb6b1f0a5db7.png
      • DONE \(P\)

        from kinetic_spont_p 

        def kinetic_spont_P(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, K=K, C_i=C_i):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    P = (-((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/
         (np.sqrt(pi)*np.cbrt(3*V_d)) + 8*K*C_i*np.sqrt(pi*S_i))/(3*eta)

    return P
        • testing
          x_range = np.logspace(6,9)
boundary = kinetic_spont_P(V_d=x_range)
boundary_test(x_param='V_d', y_param='P', x_range=x_range, boundary=boundary,
              y_offset=-0.5)
          6c852af3a08c192d995a14487626a0d5308a5aaf.png
      • DONE \(\sigma\)
        • testing
          x_range = np.logspace(6,9)
boundary = kinetic_spont_sigma(V_d=x_range)
boundary_test(x_param='V_d', y_param='sigma', x_range=x_range, boundary=boundary,
              y_offset=-0.5)

          ---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-122-68222cac32b7> in <module>
      1 x_range = np.logspace(6,9)
----> 2 boundary = kinetic_spont_sigma(V_d=x_range)
      3 boundary_test(x_param='V_d', y_param='sigma', x_range=x_range, boundary=boundary,
      4               y_offset=-0.5)

<ipython-input-121-594c148ecf9a> in kinetic_spont_sigma(P, gamma_m, gamma_d, V_d, S_i, K, C_i)
      7     eta = (S_i**(3/2))/(6*pi**(1/2))
      8 
----> 9     sigma = (( np.sqrt(pi)/S_i**(3/2) )*np.cbrt(3*V_d)(8*K*C_i*np.sqrt(pi*S_i)-3*P*eta)-
     10              gamma_d*epsilon+gamma_m*beta)/(alpha+beta)
     11 

TypeError: 'numpy.ndarray' object is not callable

          :END:

      • DONE \(C_{i}\)

        from kinetic_spont_ci 

        def kinetic_spont_C_i(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  V_d=V_d, S_i=S_i, K=K, P=P):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    C_i = (((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/
         (np.sqrt(pi)*np.cbrt(3*V_d)) + 3*P*eta)/(8*K*np.sqrt(pi*S_i))

    return C_i
        • testing
          x_range = np.logspace(6,9)
boundary = kinetic_spont_C_i(V_d=x_range)
boundary_test(x_param='V_d', y_param='C_i', x_range=x_range, boundary=boundary,
              y_offset=0.5)
          8ae3683cfcd331d06dba70c3768ada1f31bc5082.png
      • CANCELLED \(\gamma_m\)
      • CANCELLED \(\gamma_d\)
      • CANCELLED \(S_{i}\)
      • DONE \(V_{d}\)

        from kinetic_spont_ci 

        def kinetic_spont_V_d(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d,
                  C_i=C_i, S_i=S_i, K=K, P=P):

    alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4))
    epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d)))
    eta = (S_i**(3/2))/(6*pi**(1/2))

    V_d = (1/3)*(((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/
         ((8*K*C_i*np.sqrt(pi*S_i)-3*P*eta)*np.sqrt(pi)))**3

    return V_d
        • testing
          x_range = np.linspace(0.0002, 0.002)
boundary = kinetic_spont_V_d(P=x_range)
boundary_test(x_param='P', y_param='V_d', x_range=x_range, boundary=boundary,
              y_offset=-5, x_scale='linear')
          47bdfd0389f80e357a6f2084e80dbce6819748db.png
    • side-by-side F plots with both phase boundaries
      def both_boundary_test(x_param, y_param, x_range, thermo_boundary, kinetic_boundary,
                       params=params, factors=[0.7,1,1.3], y_offset=0.2, x_scale='log', y_scale='linear',
                       variable_df=variable_df):
    fparams = params.copy()
    fig, (ax1, ax2)  = plt.subplots(nrows=1, ncols=2, figsize=(10,4))


    # if x_scale == 'log':
    #     ax1.semilogx(x_range, thermo_boundary, color='black', label='thermodynamic')
    #     ax1.semilogx(x_range, kinetic_boundary, color='blue', label='kinetic')
    # # else:
    #     ax1.plot(x_range, boundary, color='black')

    if y_offset > 0:
        fill_range = np.max(kinetic_boundary)
    if y_offset < 0:
        fill_range = np.min(kinetic_boundary)

    ax1.loglog(x_range, thermo_boundary, color='yellow')
    ax1.fill_between(x_range, y1=thermo_boundary, y2=fill_range*1.3, color='yellow', alpha=0.8, label='globally stable')
    ax1.loglog(x_range, kinetic_boundary, color='blue')
    ax1.fill_between(x_range, y1=kinetic_boundary, y2=fill_range*1.3, color='blue', alpha=0.8, label='locally stable')

    ax1.legend()

    for i, factor in enumerate(factors):
        middle = len(x_range)//2
        x_pick = x_range[middle]*factor
        for j, boundary in enumerate([thermo_boundary, kinetic_boundary]):
          boundary_pick = boundary[middle]+(factor-1)*np.sqrt(thermo_boundary[middle]**2)*y_offset
          fparams[x_param] = x_pick
          fparams[y_param] = boundary_pick
          F = llps_model_recbrt(h=fparams['h_range'], sigma=fparams['sigma'], A_t=fparams['A_t'],
                  S_i=fparams['S_i'], K=fparams['K'], C_i=fparams['C_i'], gamma_d=fparams['gamma_d'],
                  gamma_m=fparams['gamma_m'], gamma_i=fparams['gamma_i'], V_d=fparams['V_d'], P=fparams['P'])
          delta_F = F-F[0]

          # ax1.scatter(x_pick, boundary_pick, linewidths=5, color=((j)/2, (j)/2, (j+1)/2, (i+1)/len(factors)))
          # ax2.plot(h_range,delta_F, color=((j)/2, (j)/2, (j+1)/2, (i+1)/len(factors)))

          ax1.scatter(x_pick, boundary_pick, linewidths=5)
          ax2.plot(h_range,delta_F)

    ax2.hlines(0, -0.05,1.05, color='black')

    ax1.set(xlabel=variable_df.loc[x_param]['description']+' ('+variable_df.loc[x_param]['units']+')',
            ylabel=variable_df.loc[y_param]['description']+' ('+variable_df.loc[y_param]['units']+')')
#            xscale=x_scale)
    ax1.set_xscale(x_scale)
    ax1.set_yscale(y_scale)
    ax2.set(xlabel='h: relative invagination height', ylabel='∆F ($pN \cdot nm$)')
    plt.tight_layout()

result plots [8/9]

  • plot all outputs vs. invagination height for default parameters
    • as individual plots
      outputs = llps_model()
outputs.pop('theta_d')
# outputs.pop('2÷R_i')
variables = list(outputs.keys())

for variable in variables:
    plt.figure()
    plot_simulation(output=variable)
    plt.xlabel('h: invagination distance/vesicle diameter')
    plt.savefig(fig_dir+'%s-vs-H_i.png' %(variable))

      2bcaae2bb70a5f6d3cd7d9f720886f53b7b098c3.png 73c40284dde2b788ac50ad17304c206f2ff8ab53.png 23585bd7388e3fcf885463524dff95c8e6ab90ef.png 235f6e3ec56b8727b27b231ca61d469bfe69a045.png 9b228be03256322a7722603cd0805be1b513b49a.png ee217b67de3fd3b64ecffbc147e5c55225fdd2f0.png 9821011cd5fec27581547a1442359ee46695f6d0.png ece5def6bfaf3645baba4bd83d0db58276ef9ef7.png f06824a89d9825bba76ca35a36df6cf1c3f44a5a.png 461dee778bc57c10ff6634bb2e689c63d1e16cc9.png d41e887ffbdcb264344c497b92858dccf65f41f7.png 2a03d989e212796d2f797f8225a83f9a9c55a3bc.png 976c5ed2ff7fedb7473e31700846fe0a20893afe.png 89cdd1bb4f97813c63ae76202d4b9f4f1a907b93.png aad1055a51032dbefe2b046636cd98c3338f9b11.png 9f1fba17ca9725f2e5f08ce2df70da8ee5cb810a.png d089aafb204d59c5c47ab38a127a41c229181911.png

    • as subplots of one figure
      outputs = llps_model()
outputs.pop('theta_d')
variables = list(outputs.keys())
nplots = len(variables)
width = 3
height = nplots//3+1
plt.figure(figsize=(width*5,height*4))

for i, variable in enumerate(variables):
    plt.subplot(height, width, i+1)
    plot_simulation(output=variable)
    plt.xlabel('h: invagination distance/vesicle diameter')

plt.tight_layout()
plt.savefig(fig_dir+'all_outputs.png')
      3f9e8f97c5cee3d83f97928599a1e69e2e3a4707.png
  • change in all outputs vs. invagination height for default parameters
    • as individual plots
      outputs = llps_model()
outputs.pop('theta_d')
variables = outputs.keys()

for variable in variables:
    plt.figure()
    plot_simulation_delta(output=variable)
    plt.tight_layout()
    plt.savefig(fig_dir+'%s-vs-H_i_delta.png' %(variable))

      0a5128988c1cc0fd1077e67cecce7bc85198afea.png 8aab118534e41198504a7b16104a7ea1a9202d4e.png 2b08afa1fd9a9ab0c8e4c62d75b4c91691d9cd9a.png a5381559e37627e5d3b1a22d9dff648b71b98229.png 91f06409b34bc48e22c0ca48dc458231eff066d2.png 5e806ce2c0adde222a5547a1f66aade5fa0e4f5e.png f5f1c9a169c4aeac506a52200f3f2c3308cc0dfb.png da63f6e19f7d481c2659f9caaa13f6ef6bafc576.png 184eedb0e71c1a55085ccb32bec4863311054dfa.png 6a9aae755bfe2b178d0a31e6c1871c8806ae1fb0.png a06f1a5d6aa86e5c88d194d79b10bbca450e031c.png 99cb6d32331bb3d1cb2944d41214623d7a8d857c.png 195a1f76d402aaa2091ad27d2f4193e789a7ea03.png 1f4e4c03ce722a09b083fc719afd43ec961d975e.png 81eab990b18dac5aab5efaff7ba7cedf48624ad4.png 7a9e81de9d2d73b442a332ea306052226e8411cc.png b976297d7731cfc494beef6fa9fd085edcdc2f7a.png

    • as subplots of one figure
      outputs = llps_model()
outputs.pop('theta_d')
variables = outputs.keys()
nplots = len(variables)
width = 3
height = nplots//3+1
plt.figure(figsize=(width*5,height*4))

for i, variable in enumerate(variables):
    plt.subplot(height, width, i+1)
    plot_simulation_delta(output=variable)

plt.tight_layout()
plt.savefig(fig_dir+'all_outputs_delta.png')
      80e2d75f6548b3e4200bcbc29ea753bbf8776e60.png
  • change in surface areas
    variables = ['S_o', 'S_d', 'S_m']

for variable in variables:
    plot_simulation_delta(output=variable, label=variable)
    plt.ylabel('∆ surface area')
    plt.legend()
    plt.tight_layout()
    plt.savefig(fig_dir+'areas-vs-H_i_delta.png')
    e9c2b8a3b5c9afcae737f31318e42c5a36f72999.png
  • each individual energy term output
    • set temporary variables
      P_temp = 1e-3
gamma_d_temp = gamma_d*10
gamma_m_temp = gamma_m*10
K_temp = 160
    • normal droplet
      show_geometry(H_i=100)
plt.savefig(fig_dir+'normal_droplet.png')
      dc1240c7e73de4aaa6b4c168b30fcd0ff3844e07.png
      • with appropriate signs

        droplet only: 

        variables = ['surface_tension', 'wetting_energy', 'droplet_energy']
colors = ['blue', 'green', 'black']

plt.figure(figsize=(6,6))

for variable, color in zip(variables, colors):
    plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp,
                          gamma_m=gamma_m_temp, K=K_temp, color=color)

plt.legend()
#plt.ylim(-100, 100)
plt.ylabel('∆ energy (pN * nm)')
#plt.title(title)
plt.tight_layout()
plt.savefig(fig_dir+'droplet_energy_contributions.png')
plt.savefig(fig_dir+'droplet_energy_contributions.svg')
        561066d087aa8261d884357ea4b340fdc0baedbd.png

        all contributions: 

        variables = ['surface_tension', 'wetting_energy', 'droplet_energy',
             'membrane_tension', 'bending_energy', 'turgor_pressure', 'F']
colors = ['blue', 'green', 'black',
          'red', 'purple', 'brown', 'orange']

plt.figure(figsize=(6,6))

for variable, color in zip(variables, colors):
    plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp,
                          gamma_m=gamma_m_temp, K=50, color=color)

plt.legend()
#plt.ylim(-100, 100)
plt.ylabel('∆ energy (pN * nm)')
#plt.title(title)
plt.tight_layout()
plt.savefig(fig_dir+'energy_contributions.png')
plt.savefig(fig_dir+'energy_contributions.svg')
        ca5adb4bc7e39e5d3c954ac2ba3192da06fd096f.png
      • flipped sign of negative energy contributions
        variables = ['membrane_tension', 'bending_energy_flipped','surface_tension',
              'wetting_energy_flipped', 'turgor_pressure']

plt.figure(figsize=(10,10))

for variable in variables:
    plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp,
                          gamma_m=gamma_m_temp, K=K_temp)
plt.legend()
#plt.ylim(-100, 100)
plt.ylabel('∆ energy (pN * nm)')
#plt.title(title)
plt.tight_layout()
plt.savefig(fig_dir+'energy_contributions_flipped.png')
        006bb289748fa11eff88e653126657a42086c598.png
    • huge droplet
      show_geometry(H_i=100, V_d=V_d*10**4,
              left=-3600, right=3600, bottom=-2300)
plt.savefig(fig_dir+'bigdrop.png')
      d548ff9962ce257e7f0d4d78e1fad4e9a6c1cc09.png
      • with appropriate signs
        variables = ['surface_tension', 'wetting_energy', 'droplet_energy']
colors = ['blue', 'green', 'black']

plt.figure(figsize=(6,6))

for variable, color in zip(variables, colors):
    plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp,
                          gamma_m=gamma_m_temp, K=K_temp, V_d=V_d*10**4, color=color)
plt.legend()
#plt.ylim(-100, 100)
plt.ylabel('∆ energy (pN * nm)')
#plt.title(title)
plt.tight_layout()
plt.savefig(fig_dir+'droplet_energy_contributions_bigdrop.png')
plt.savefig(fig_dir+'droplet_energy_contributions_bigdrop.svg')
        db3e36d09f5fdc9679496495cbdb0531eab8ed26.png
      • flipped sign of negative energy contributions
        variables = ['membrane_tension', 'bending_energy_flipped','surface_tension',
              'wetting_energy_flipped', 'turgor_pressure']

plt.figure(figsize=(10,10))

for variable in variables:
    plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp,
                          gamma_m=gamma_m_temp, K=K_temp, V_d=V_d*10**4)
plt.legend()
#plt.ylim(-100, 100)
plt.ylabel('∆ energy (pN * nm)')
#plt.title(title)
plt.tight_layout()
plt.savefig(fig_dir+'energy_contributions_bigdrop_flipped.png')
        6cc2bad9a6422e5faa6fc6d36227db442a33a407.png
  • Plot F for a series of values of gamma_m

    What are the parameter values? I assume that gamma_m is negative. Plot F for a series of values of gamma_m. Based on the plots for S_m, S_d, and S_0, F should end up with a minimum at H/R=1 if gamma_m is large enough. It might also happen if gamma_d is small enough. It's hard to say because I can't easily compare the magnitude of changes in S_d, S_i, and S_0.

    • absolute
      gamma_m_range = [0,50,100,150,199]

for i in gamma_m_range:
    plot_simulation(gamma_m=i, label=i/gamma_d)

plt.legend(title='gamma_m/gamma_d')
plt.xlabel('invagination distance/vesicle diameter')
plt.tight_layout()
plt.savefig(fig_dir+'f_scan_gamma_m.png')

      <ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
      251d320215ce964a1d23ef1abd8347adddcd9873.png
    • normalized
      gamma_m_range = [0,50,100,150,199]

for i in gamma_m_range:
    plot_simulation_normalized(gamma_m=i, label=i/gamma_d)

plt.legend(title='gamma_m/gamma_d')
plt.xlabel('invagination distance/vesicle diameter')
plt.tight_layout()
plt.savefig(fig_dir+'f_scan_gamma_m.png')

      <ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
      6e570113298eaeacf47b71d75a671717c9eb4e2e.png

      :end:

    • delta
      gamma_m_range = np.array( [0.01,0.25,0.5,0.75] )*gamma_d

for i in gamma_m_range:
    plot_simulation_delta(gamma_m=i, label=i/gamma_d)

plt.legend(title='$\gamma_{m}/\gamma_{d}$')
plt.tight_layout()
plt.savefig(fig_dir+'f_scan_gamma_m_delta.png')
      a60b2c02931b7f50512a91322d36e23bd0489771.png
    • delta and geometry side by side
      gamma_m_range = [0,50,100,150,180]

for i in gamma_m_range:
    plot_simulation_geometry(gamma_m=i, label=i/gamma_d)
    plt.legend(title='$\gamma_{m}/\gamma_{d}$')
    plt.tight_layout()
    plt.savefig(fig_dir+'f_scan_gamma_m_%s.png' %(i))

      <ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
/usr/local/Caskroom/miniconda/base/envs/llps_model_analysis/lib/python3.9/site-packages/numpy/core/_asarray.py:83: UserWarning: Warning: converting a masked element to nan.
  return array(a, dtype, copy=False, order=order)
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
/usr/local/Caskroom/miniconda/base/envs/llps_model_analysis/lib/python3.9/site-packages/numpy/core/_asarray.py:83: UserWarning: Warning: converting a masked element to nan.
  return array(a, dtype, copy=False, order=order)
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
/usr/local/Caskroom/miniconda/base/envs/llps_model_analysis/lib/python3.9/site-packages/numpy/core/_asarray.py:83: UserWarning: Warning: converting a masked element to nan.
  return array(a, dtype, copy=False, order=order)
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8
/usr/local/Caskroom/miniconda/base/envs/llps_model_analysis/lib/python3.9/site-packages/numpy/core/_asarray.py:83: UserWarning: Warning: converting a masked element to nan.
  return array(a, dtype, copy=False, order=order)
<ipython-input-7-3ad17b48a703>:6: RuntimeWarning: invalid value encountered in arccos
  theta_d = np.arccos(gamma_m/gamma_d) #equation 8

      6462d186e7e67b48545aa7e42a964814ae14beca.png 47486545ffb5778b7bb4d466f146394f948ae119.png 3a4f99dfba1466623492d07043b624ba8d1c681a.png a2d43c7184c60ce612ac1f497d7ea473ec383a51.png 28c26346adc7c9f4a7e2c794c84fe02b222f5741.png

  • F over series of turgor pressure
    pressure_range = np.linspace(10**-2,1,5)

for i in pressure_range:
    plot_simulation_delta(P=i, label=i)

plt.legend(title='turgor pressure ($N/nm^{2}$)')
plt.tight_layout()
plt.savefig(fig_dir+'f_scan_turgor_delta.png')
    d9ce597d421ffa065f6d13e6643381e174b76c1c.png
  • spontaneity analysis

    there's no solution for kinetic \(\sigma\)

    • DONE \(V_{d}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2),
              #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(7,9,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(V_d=x_range)
    kinetic_boundary = kinetic_boundaryy(V_d=x_range)
    both_boundary_test(x_param='V_d', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/2, factors=[0.9,1,1.1])
    plt.savefig(fig_dir+y_param+'V_d_boundary_check.png')

      65ba6117df9694fcbad30a1046dfecdb2c257d41.png fa88cd1ae50969bfacee7b577b140af204b654f9.png 567102d0d44ac46c20f4b184996553f276ad06bd.png 

      show_geometry(H_i=100, V_d=10**(7))
plt.savefig(fig_dir+'10e7_geometry.png')
      89e844cc38a76604a4a2bb5514d8aff005cef79b.png 
      show_geometry(H_i=100, V_d=10**(9), left=-1200, right=1200, bottom=-600)
plt.savefig(fig_dir+'10e9_geometry.png')
      dbb38358797f6b2b639bb1379d44b169578b65b3.png
    • DONE \(\gamma_{d}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2),
              #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(-4,-1,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(gamma_d=x_range)
    kinetic_boundary = kinetic_boundaryy(gamma_d=x_range)
    both_boundary_test(x_param='gamma_d', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/2, factors=[0.9,1,1.1])
    plt.savefig(fig_dir+y_param+'gamma_d_boundary_check.png')

      6cc97f36dd06b88d6cebab05fd7138c2fabec7e4.png 984e126283b2f9070d6ba6d80ded5332647b18be.png bb83131b8242be061b1aba8673800328c56e8428.png

    • DONE \(\gamma_{m}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2),
              #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(-4,-1,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(gamma_m=x_range)
    kinetic_boundary = kinetic_boundaryy(gamma_m=x_range)
    both_boundary_test(x_param='gamma_m', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset*2, factors=[0.8,1,1.2])
    plt.savefig(fig_dir+y_param+'gamma_m_boundary_check.png')

      2767bdbac987199f7030d43be11dc7cdabac7cf3.png ba2373671cc8500e5fcc09ab8261f847b4a8cf45.png 715e736e02880dd6755fdbdcc886399d7112a7e5.png

    • TODO \(\gamma_{m}\) /\(\gamma_{d}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2),
              #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(-2,0.0001,base=10)*gamma_d
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(gamma_m=x_range)
    kinetic_boundary = kinetic_boundaryy(gamma_m=x_range)
    both_boundary_test(x_param='gamma_m', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/2, factors=[0.8,1,1.2])
    plt.savefig(fig_dir+y_param+'gamma_m_relative_boundary_check.png')

      258ac3c3dc1ee5116e9f854e6110318b8b3a4f72.png 56e5e90eb06ab1caa3b23cb2866776a606f4c46a.png 6058ac26b1ef2f42ca5fc11f0166204e4072c412.png

    • DONE \(\sigma\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', -2),
              (thermo_spont_P,kinetic_spont_P, 'P', 2),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', -2)]
x_range = np.logspace(-5,-1,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(sigma=x_range)
    kinetic_boundary = kinetic_boundaryy(sigma=x_range)
    both_boundary_test(x_param='sigma', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/5, factors=[0.8,1,1.2])
    plt.savefig(fig_dir+y_param+'sigma_boundary_check.png')

      0cc8731998e4ad151c85f1fd420f6314b90cdd53.png 5a5f45ca0b6ece0d454b27b7ef64856959fa9538.png a916aa303d51ec1b24462831f1d620c826d3c978.png

    • DONE \(S_{i}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(3.5,4.5,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(S_i=x_range)
    kinetic_boundary = kinetic_boundaryy(S_i=x_range)
    both_boundary_test(x_param='S_i', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset*50, factors=[0.999,1,1.001])
    plt.savefig(fig_dir+y_param+'S_i_boundary_check.png')

      c5f11526b60a9b6da37178de779f0cec2fae9c52.png 9f61ad6cea08c32b2f3a9aa858dcc20af2edd33c.png f2cd6de8326702e0be621e11ec81b62d2fd1bcd7.png

    • DONE \(P\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              #(thermo_spont_P,kinetic_spont_P, 'P', 2),
              (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)]
x_range = np.logspace(-3,0,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(P=x_range)
    kinetic_boundary = kinetic_boundaryy(P=x_range)
    both_boundary_test(x_param='P', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset*150, factors=[0.999,1,1.001])
    plt.savefig(fig_dir+y_param+'P_boundary_check.png')

      87fb4db718f01e5714b1ad4006cf2488cf99d621.png 452e33f681d06561a1dcb3c35ca0c43de50c6009.png

    • DONE \(C_{i}\) x-axis
      boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2)]
x_range = np.logspace(-4,-1,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(C_i=x_range)
    kinetic_boundary = kinetic_boundaryy(C_i=x_range)
    both_boundary_test(x_param='C_i', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/5, factors=[0.8,1,1.2])
    plt.savefig(fig_dir+y_param+'C_i_boundary_check.png')

      0156dbd6ed414abd7ba08685c859e2c5840ea784.png ed11232b216463c405702c2d707f99b85b33f232.png

    • DONE \(K\) x-axis
      boundaries = [(thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2),
              (thermo_spont_P,kinetic_spont_P, 'P', -2)]
x_range = np.logspace(1,4,base=10)
for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries:
    thermo_boundary = thermo_boundaryy(K=x_range)
    kinetic_boundary = kinetic_boundaryy(K=x_range)
    both_boundary_test(x_param='K', y_param=y_param, x_range=x_range,
                    thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary,
                    y_offset=y_offset/1, factors=[0.9,1,1.1])
    plt.savefig(fig_dir+y_param+'K_boundary_check.png')

      8b69a908af8af6ca2d3dae47be3a54ca31b88776.png 3e4342aa08761e2dc11b17a500cc7fcd808dba9a.png

  • plots for 20200915 lab meeting drafting
    • TODO each individual energy term output
      variables = ['membrane_tension',
              'bending_energy','surface_tension',
              'wetting_energy','turgor_pressure']

for variable in variables:
    plt.figure()
    plot_simulation_delta(output=variable)
    plt.tight_layout()
    plt.savefig(fig_dir+'%s-vs-H_i_delta.png' %(variable))
      67951faa63202a353880478d30043157eeaa86be.png
    • membrane components without droplet
      • effect of membrane tension alone
        plot_simulation_geometry(V_d=0, K=0, gamma_m=0, gamma_i=0, P=0)
        33f0c1280edf5be0ec62ca4e853e68b028415410.png
      • effect of bending energy alone
        plot_simulation_geometry(K=K, V_d=0, sigma=0, gamma_m=0, gamma_i=0, P=0)
        96eea197f56156d4bfe2d2ffce1cdbc62280f3f0.png
        • TODO fix this

          why is this different from plotting membrane bending energy term alone? 

          R_i_range = np.linspace(0.1,np.pi,100)
bending_energy = (K/2)*S_i*((2/R_i_range)-C_i)**2
plt.plot(R_i_range, bending_energy)
          <matplotlib.lines.Line2D at 0x159e3c430>
          837e85a1c8f421f9d96cbd477b15d0bf46cff805.png
    • droplet effects
      • surface tension
        plot_simulation_geometry(sigma=0, K=0, gamma_m=0, gamma_i=0, P=0)
        c9a52aa61350b6e13e5bc5001b7b3fdef06b2809.png
      • surface tension + wetting energy

        can't have wetting energy alone 

        plot_simulation_geometry(sigma=0, K=0, P=0)
        c147c16a1be0becdcede197eef996043f8488b6c.png
    • TODO droplet+membrane effects
    • TODO turgor pressure effects

Author: Max Ferrin

Created: 2023-01-04 Wed 18:05

Validate